{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "ad7a960b-6903-44f9-b6bf-ebeed61dfd2b",
   "metadata": {},
   "source": [
    "**泊松分布**是一种离散概率分布，用于描述在一定时间或空间内，某事件发生的次数。当以下条件满足时，事件通常可以用泊松分布来描述：\n",
    "\n",
    "- 事件是独立发生的：在给定的时间或空间范围内，一个事件的发生与否不会影响其他事件的发生概率。例如，在某十字路口，每一辆车的到达都是相互独立的，一辆车的到达不会影响其他车到达的概率，这种情况下车辆到达的事件就满足独立性条件。\n",
    "- 事件发生的概率是恒定的：在任意相等的时间或空间间隔内，事件发生的概率是固定不变的。如在生产线上，每单位时间内出现次品的概率如果大致相同，那么次品出现的次数就可能符合泊松分布。\n",
    "- 在短时间或小空间内，事件发生的概率与时间或空间的长度成正比：比如在一定时间内，放射性物质发射出的粒子数，在很短的时间间隔内，发射出粒子的概率与时间间隔的长度近似成正比，且在互不重叠的时间间隔内，粒子发射的次数相互独立，这种情况就满足泊松分布的条件。\n",
    "- 在相同的时间或空间范围内，事件发生的次数可以无限细分：理论上，事件发生的次数可以是任意非负整数，不存在一个最小的、不可再分的发生次数单位。像在一段时间内，网站的访问量，从理论上来说可以是 0 次、1 次、2 次…… 无穷多次，并且可以无限细分时间来统计访问次数，这也符合泊松分布的特征。\n",
    "\n",
    "常见的符合泊松分布的例子有：某医院在一定时间内前来就诊的患者人数、某地区每月发生的交通事故次数、一本书中每页的印刷错误个数等。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "987f6933-d3a2-4aa1-9c43-c57946680707",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "Call:\n",
       "lm(formula = weight ~ height, data = women)\n",
       "\n",
       "Residuals:\n",
       "    Min      1Q  Median      3Q     Max \n",
       "-1.7333 -1.1333 -0.3833  0.7417  3.1167 \n",
       "\n",
       "Coefficients:\n",
       "             Estimate Std. Error t value Pr(>|t|)    \n",
       "(Intercept) -87.51667    5.93694  -14.74 1.71e-09 ***\n",
       "height        3.45000    0.09114   37.85 1.09e-14 ***\n",
       "---\n",
       "Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1\n",
       "\n",
       "Residual standard error: 1.525 on 13 degrees of freedom\n",
       "Multiple R-squared:  0.991,\tAdjusted R-squared:  0.9903 \n",
       "F-statistic:  1433 on 1 and 13 DF,  p-value: 1.091e-14\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fit <- lm(weight~height,data=women)\n",
    "summary(fit)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "67fd409c-bb43-43ce-8f64-57600e528f16",
   "metadata": {},
   "source": [
    "**Residual standard error(残差标准误差)**\n",
    "\n",
    "单元线性回归中：$$RSE = \\sqrt{\\frac{\\sum(y_i-\\hat y_i)^2}{n-2}}$$\n",
    "p元回归中：$$RSE = \\sqrt{\\frac{\\sum(y_i-\\hat y_i)^2}{n-p-1}}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "28fb02de-cd0e-41c5-9b18-55fa978267d3",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "1.52500525429949"
      ],
      "text/latex": [
       "1.52500525429949"
      ],
      "text/markdown": [
       "1.52500525429949"
      ],
      "text/plain": [
       "[1] 1.525005"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "y <- women$weight\n",
    "y_hat <- fitted(fit)\n",
    "sqrt(sum((y-y_hat)^2)/(length(y)-2))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7f54f3a3-fb48-403c-8862-f5e95c5b089f",
   "metadata": {},
   "source": [
    "**Multiple R-squared:  0.991:** 表明模型可以解释体重99.1%的方差，它也是实际和预测值之间相关系数的平方($R^2 = r^2_{\\hat y y}$)。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "5271a358-2bfd-422a-ba17-2e71d8371774",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "15"
      ],
      "text/latex": [
       "15"
      ],
      "text/markdown": [
       "15"
      ],
      "text/plain": [
       "[1] 15"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "nrow(women)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "57ab6aa7-0d8d-4544-85c0-2aec2f7064bf",
   "metadata": {},
   "source": [
    "**F-statistic(F统计量)：** 线性关系是否显著的检验?$$F = \\frac{SSR/k}{SSE/n-k-1} = \\frac{MSR}{MSE}\\sim F(k,n-k-1)$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "916169bf-beae-4821-86fd-42c208d86925",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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mMonwdPnc6s3SwLgrQzdT9Kx9rsFGvU\nonzOvYkMx5FmWPLW7liacntow9TsN+X1s0a+1gqL0OMLLdtH+nvcqWUl2xydKKtzdPhSSycb\n9nV7/dTy253DZqKubtHfi3FmA+huAYIEeltA9QnZYvQ2LXNRWofobAVRkI58jCJW9LXEouNI\nfSvPa4WfcKhIZMkWqX+X8pV02o7qOkJHqzi/itAk1NcN+lmGWbuM0c06BClf9LLQ0iBtZ1z7\nezpK7ACdrOT22t9TUWP76GMpx9f+nogi28a0txizdlmig9UWBqmef+3vKaizXfSv3NLJhuqH\na39/R6Gtonv1Fl2Oy9pN9qi0TfSuBQQpO3SuDRyQzQ19awVBygxda4futi7lxIs/TkG1LeHw\nkS26IBmj+5As5baDfrVm6Vu7ddFehmtXmP2pmnpB4u8ouBV0qz2LD8herkF8MOWp0X1Klorb\nQK9apDpF6HKPF8Uada2pGsIDnWrT4pNWb1ukgiCFjT61avFbu9s+Un36M1Nvf2RjrTCOLrVr\n6WRDeZv8bjdIso9UUHU1etSyxQdkd9U5RlW7WTIbzSqdKLsah4+s48yGDNCd9hGk9NGbDiw6\n+/s0667mdtcKn9CZLhCk1NGXTvDWLnF0pRsEKW30pCOS6e/TqdLdiLlF+TWY9nZGckD23Ewh\nTRL1l3jpRvGuLB4WBmlryqYtztasZat0Ikgaz73YpYgo2bH4pNXGRn0otsDr9uj9t6Eg+BgF\nQQrShxzRuXYsDNLqukU6cOvLwLx2IUGySbOPtBNfTJ9aL/WmBwmSTYsvWfz4GIUQtV7obQey\nj2SR6GMUf6LVuaLYi3zYYWXWziLObEjP597jOJI1C4JUVNu9lZu6EKRF6Dwfll5Ev6g2O+3p\nQV3T8hbzQd95sSBIzX67vt6Kuar/Dp7XChd0nR+L95EO23XJ55GCQc95opls2JUEKQh0nC+C\nIO03K7ZIQWBOzp+FQTpuuyOyq432TrIMiF/Qax4tmWzY1UU7b7feySfBGRI/oNN8Wjj9XW31\nc98nxsQv6DOvFgWptnQ8lkExH13mF1ukNNBjni3aR1qzjxQIOsw3Zu1SQH95x3Gk+HH4KACc\n2RA9OisEnGsXO/oqCJz9HTm6Kgx8Hilu9FQg+IRs1OioUHDNhpjRT8EgSPFi2jsgBCladFJI\nCFKs6KOgEKRI0UVhIUhxoocCQ5CiRAeFhiDFqNc/XIU4DAQpQo/u4br4oSBI0ennhju1hIIg\nxca8Waa/vCNIkTHvvqC/vCNIcTFvv6K/vCNIUXnuGPaRQkGQYvLSL8zahYIgReRdt3AcKQwE\nKRokJmQEKRb0SdAIUiTokrARpDjQI4FzGqT9prpevevLFY4ZNk/okNA5DFJzvQhep1SvVdLo\nj+A5DFJtiutlJI+7wtRjD2XgDNAd4XMYpMI8rsZ6MMXYQxk5ffRGBBwGaXAcZPygCEPngcNH\nUWCLFDi6Ig5u95GulwlnH2kyeiISLqe/y96s3Wr0quEMnys6IhZujyPV1fUOFhxHmoR+iAZn\nNgSMbogHQQoXvRARD0HaFma1HX8IQ4hp78i4DNKhMsX2tOEUoSnogrg4DNKhS1Bt1s3pWJnR\nbRKjiB6IjMMgrdtjR/XlSGxjVmMPzX4YZd8B0XF+ipCpel8Mf9zz41OkIvfXHyHnQfq7vKfj\nFKERmb/8KDl9a7e+nc7QrDlF6LO8X32kXH6wr7i/ZTPjG6S8h1LWLz5aTo8j1bf4FKPbo6zH\nUvb7h5HizIaw5PvKI0eQgpLtC48eQQrJ5QAB7+4iRJACcr8iPlGKDkEKh7n/L9MOiJnTA7KT\nT17Ichz1cpRpD8TMYZC2BGmEGW6KMuyBuDn9GEUx/uGJh/yG0XOA8uuByDndRzqMnxj0kN0w\nMk9L2XVA9NxONmx7l7Ybk9s46r1eZu3ixKxdAIYvl+NIMSJI/uX1ahNFkLzL6sUmiyD5ltNr\nTRhB8ov9oUQQJK+yeaHJI0g+5fI6M0CQPMrkZWaBIPmTx6vMBEHyJosXmQ2C5EsOrzEjBMkP\npr0TQ5C8SP4FZocg+ZD668sQQfIg8ZeXJYLkXtqvLlMEybmkX1y2CJJrKb+2jBEkxxJ+aVkj\nSE5x+ChVBMmlVF8XCJJLib4snAiSS2m+KnQIkjNJvihcESRXUnxNuCNIjiT4ktBDkJxg2jt1\nBMkBLkKcPoJkXRcjopQ4gmSdGdyKD2kiSLbdtkUpvSa8IEi2cQ++LBAkywxBygJBsstwM8s8\nECSbHvexZNYucQTJovu7OmKUPIJkTxqvApMQJGuSeBGYiCDZksJrwGQEyZIEXgJmIEh2xP8K\nMAtBsiL6F4CZCJIFzHbnhyDpxb32+AlBkot65fEjgqQW87rjZwRJLOJVxwIESSveNcciBEkq\n2hXHQgRJiGnvfBEknTjXGhIESSbKlYYIQVKJcZ0hQ5BEIlxlCBEkjfjWGFIESSK6FYYYQVKI\nbX0hR5CW4/ARCNJyUa0sLCFIS8W0rrCGIC0U0arCIoK0TDxrCqsI0iLRrCgsI0i/uVzOO/z1\nhCME6RfXG0yEvppwhyD94nJX2NDXEg4RpB/c7q4c+GrCIYL0A9P7D2gRpB+Yx0YJ6BCkXxhy\nhCGC9AtuC4snBOkHhrlvPCFIsxEhvCJIc4W8bvCGIM0U8KrBI4I0T7hrBq8I0izBrhg8I0hz\nhLpe8I4gzRDoaiEABGkypr3xGUGaKsR1QjAI0kQBrhICQpCmCW+NEBSCNElwK4TAEKQpQlsf\nBIcgTRDY6iBABOm7sNYGQXIapP2mMq2q3o8/MKShy+EjTOAwSM3KPJTqtbIloFVBwBwGqTbF\n36FbOu4KU489NJzRG86aIGgOg1SYw335YIqxhwYzfINZEQTOYZAGOxvjex6hjN9Q1gPBY4s0\nIpDVQATc7iPtjt1SJPtIYawFouBy+rvszdqtGvFayTHtjRncHkequ+NIRbUJ/zhSAKuAiHBm\nQ6hrgKgQpDBXAJFxGaRmbUy5uz5vcNPf/YsQkyPM5PIUoeJyot3leQML0uCy+OQIczmd/t6e\n07QtutPsggvSy/+BGZwekO3+ORarY3BBMr1/yRHm83CKUFOW74Jk+n58ip89gsThI/zCYZBW\n5nYQdlUGu0UiRviJwyBtzfq6dDRlYEG67R2RI/zG5fR3fR+nuy/v3jzP2gFzOT0ge6huS8d1\nYEG67KN5eFqkgTMbfD8rkkCQ/D4pEkGQbk9JjrAAQfL1jEgKQfLzhEiM0zMbJp+84HpckyMs\n5PSAbKhBIkdYyuVbu0Mxfn3VB7cjmxxhMbcHZMevHfTgdGiTIyzndrJh27u03RiXY5scQSD3\nWTsOH0Ei8yARI2jkHSRyBJGsg0SOoJJzkMgRZDIOEjmCTr5BIkcQyjVITHtDKtMgESNo5Rkk\ncgSxLINEjqCWY5DIEeQyDBI5gl5+QSJHsCC7IJEj2JBZkDh8BDtSDtLrhSGIESxJN0hvLotP\njmBLwkF6aYkcwZpkg2Se/iVHsCmfIJEjWJRNkMgRbEo2SMN9JKa9YVfCQerN2hEjWJZukHrH\nkcgRbEs5SHZaA97IIEjkCPalHyRyBAeSDxI5ggupB4kcwYm0g8ThIziSdJCIEVxJOUjkCM4k\nHCRyBHfSDRI5gkPJBokcwaVUg0SO4FSiQSJHcCvJIHH4CK7FFaTXC2zN+W3AmpiC9OYCW9N/\nGbApqiCN/Ozb7wJWRRSkaZcxIUfwIbUgkSN4kViQyBH8iChI3/eRmPaGL1EF6cusHTGCNzEF\n6ctxJHIEf+IKkvp3AJFkgkSO4FMqQSJH8CqRIJEj+JVGkMgRPEshSBw+gncJBIkYwb/4g0SO\nEIDog0SOEILYg0SOEITIg0SOEIa4g0SOEIiYg8S0N4IRcZCIEcIRb5DIEQISbZDIEUISa5DI\nEYISaZDIEcISaJCAyMwf5Q6C9JHNLZHFtlltl23H0jRBCqlpVjvapglSSE2z2tE2TZBCaprV\njrZpghRS06x2tE0TpJCaZrWjbZoghdQ0qx1t0wQppKZZ7WibJkghNc1qR9s0QQqpaVY72qYJ\nUkhNs9rRNk2QQmqa1Y62aZ9BApJBkAABggQIECRAgCABAgQJECBIgABBAgQIEiBAkAABggQI\nECRAgCABAgQJECBIgABBAgQ8BKl3nfKmLkxRN8LGD2tj1sduUd12r+ntyt5qn+2VHznrNT18\nFmnb4koOrmZvsWldId0H6fB4HceiWyp0td1dGmy7puwWVzaarh+L8rbPmkIYpF7TO3urra7k\nbbAXJ3kh+00LC+kjSNVtcW3qU/tq1rLGi+Jwaqq22b05Lx4Ks9c3fTDrc9dvrax2q1LemLrX\n9PBZpG3rK9natdWTF/LRtLKQ7oO0NZvb4nXA6MbNX1fPpv1rU5td943Nt9+Z33RlcbW7r4RB\n6jU9fBZt2/JKtpqi/ZOrLmSvaWUhfQRpe1u8voXRlXZtDrfFyrRvM3qbP13TV7pRM2j7aErh\neOw1/foKdG3LK9mqTHPSF7LX9FWkQarMbn3ew2sXN9c3BLI/Nitz2hTd9lr+N7LX9EVjSlHT\nw7ZLcxQGqdf0yysQti2v5KnNTjdIbGzsDv33t5pC+ghSp1v5bbuPWmy//cpkxlS3/Uh1//ea\nvth2bznkbW/Mn3LQDHrk6RUI25ZX8nTfatgI0mCDpCmk+yCZ80g5NXX3Bm/TZUr3Z8y0O6bN\num1RH6R7051joXuv0Wu7ewsjDVKvR4avQNm2vJLdTMD1Wfr/SJvuiArp64Bs085nbtst7LkS\nsj9kpnvXfmzb1gfp3nSrKWRv7AZtr9q5WGmQej0yeAXStuWVvE0yWAlS3dsGqQrp7cyGtmNW\n3Ra2UZb2/k8hD9KgvVJ3gKrf9rorsTRI93/kI7LXoLyS9wLKC9lruqUqpNcgyUvbm8+8TPYc\nZZM9g6nS46pUniDwaHvJDeq/NK2fta8shvQxTacu5GAGUFdI90Equj9eXcdc/jIIj2xsuj/o\nx3Ym47K4kx1+7DV9blX4vm7QtjxILz1y1K17r0F5JR9HSdSF7B+AERbSfZDq7u109zb1vNhc\nv6Fxfr/etG/V//QHxHtNC8fiS9sd4R/2wWoPn0XZtryS5w3R9SCV/syGe9PKQroPUnM5Lavr\n8vIxE66xeTS4Erf9aHot3moMVrul3B/oNb2x2NvySq7uM9TqQj6aVhbSwz5Se6Lw6rpx7U7s\nVTa+K28NNuq2702r33712z5dnkDXcr/p4bNo21b39qMP5IW8N60sJJ9HAgQIEiBAkAABggQI\nECRAgCABAgQJECBIgABBAgQIEiBAkAABggQIECRAgCABAgQJECBIgABBAgQIEiBAkAABggQI\nECRAgCABAgQJECBIgABBAgQIEiBAkAABggQIECRAgCABAgQJECBIgABBAgQIkge9W8a9//7L\nl7vbv9X7+/l9uuncl5vRVbvRH2M6guTB7CCtrkvH9u6nwiA15ji+ppiKIHnwKUifHnZfKufe\nSvXb7VFr8f3Z80WQPPg1SH/3O33PfqYPGvM3s0W8R5A8eArSdmWK7ePLujB1t3j+rzbF5nr3\n7fNPVuXtUcYcq+5HvYb63zu3UR5PvSb6T9PejNyUl92jcmX7xWaCIHkwDFLV5aS8fVm2X60v\nyeh+tL0FaW/ucTOmaL+36bfY+17XRtH0mug/zbZbujS2NXunLz1ZBMkD83DePJiyOTWl2V3i\nsDPF4XQoLsk4/2RrVvdNlTlcfrv/o1uL/e/9tUvr83at973e0xRtO3+XXz6YubtdeIsgeTAI\nUtXt+DTmOrFdtUP9POy7ZOxPt4y0v1ZedpGGP7q12P9e1S41phh+r/c0j1nvxjDdIEGQPBi8\ntetl6pGZ3uJw6d03Pjz88Yinp6nPb/gOh+d1wSIEyQPPQTpt2p2p4jhcFyxCkDx4CtLg+/aC\n1F+DXb267mARJA2C5MEgSNVjj+VlH+n2zTf7SIN2nr5X9vaR3jxNfyXYRxIhSB4MgvTXTtOd\ntrfJhsGs3e0xpjuVp75MVX8P0radoasvs3ZvnmbVHoW9ztrtmbXTIEgeDI8jdQd9ul2W3pfD\nZKxMu3nZX44bfQ9S/zjSm6f5uzxBl8oNx5E0CJIHwyC1pxyY9fH+ZXtWwn6YjP2qDdLgzIZB\nOy/fayfmHi0+Pc3lzIZLgDizQYQghentrstOfrL28XnPCT8iSIEx7Q5MU73fdZl99vc3nP2t\nQpACs7nswBRvf3icffr3OD6PJEOQQrM978CsPm15wOwTCwAAAGlJREFUdmvpc615Y6dCkAAB\nggQIECRAgCABAgQJECBIgABBAgQIEiBAkAABggQIECRAgCABAgQJECBIgABBAgQIEiBAkAAB\nggQIECRAgCABAgQJECBIgABBAgQIEiBAkAABggQIECRA4D8lKa0WciOt3wAAAABJRU5ErkJg\ngg==",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 420,
       "width": 420
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot(women$height,women$weight,xlab=\"Height(in inches)\",ylab=\"Weight(in inches)\")\n",
    "abline(fit)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "bb3c248b-8bcb-44ec-b742-80dc388538c5",
   "metadata": {},
   "outputs": [],
   "source": [
    "fit2 <- lm(weight~height+I(height^2),data=women)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "9908a72a-7404-464b-b367-525522424a42",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "Call:\n",
       "lm(formula = weight ~ height + I(height^2), data = women)\n",
       "\n",
       "Residuals:\n",
       "     Min       1Q   Median       3Q      Max \n",
       "-0.50941 -0.29611 -0.00941  0.28615  0.59706 \n",
       "\n",
       "Coefficients:\n",
       "             Estimate Std. Error t value Pr(>|t|)    \n",
       "(Intercept) 261.87818   25.19677  10.393 2.36e-07 ***\n",
       "height       -7.34832    0.77769  -9.449 6.58e-07 ***\n",
       "I(height^2)   0.08306    0.00598  13.891 9.32e-09 ***\n",
       "---\n",
       "Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1\n",
       "\n",
       "Residual standard error: 0.3841 on 12 degrees of freedom\n",
       "Multiple R-squared:  0.9995,\tAdjusted R-squared:  0.9994 \n",
       "F-statistic: 1.139e+04 on 2 and 12 DF,  p-value: < 2.2e-16\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "summary(fit2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "f2053465-74cb-47f7-a636-a09877e2922b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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R9QgLlUn5DNRm/TMhd1dIv+n00UpAsf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\nCBIgQJAAgf8BEtWuBjnQunYAAAAASUVORK5CYII=",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 420,
       "width": 420
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot(women$height,women$weight,xlab=\"Height(in inches)\",ylab=\"Weight(in inches)\")\n",
    "lines(women$height,fitted(fit2))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "221136d4-9e72-4ad7-b5e8-d957c6832aad",
   "metadata": {},
   "source": [
    "**Std.Error系数的估计标准误差**\n",
    "\n",
    "假设一元线性回归：$$\\hat y_i = \\hat\\beta_0 + \\hat\\beta_1x_i$$，要看线性系数是否显著，需要计算如下统计量：\n",
    "\n",
    "- 估计标准误差:$$s_e = \\sqrt{\\frac{\\sum(y_i - \\hat y_i)}{n-2}} = \\sqrt{MSE}$$\n",
    "\n",
    "- $\\hat\\beta_1$的估计的标准差：$$s_{\\hat\\beta_1} = \\frac{s_e}{\\sqrt{\\sum x_i^2 - \\frac{1}{n}$sum x_i)^2}}$$\n",
    "\n",
    "- t统计量$$t = \\frac{\\hat\\beta_t}{s_{\\hat\\beta_1}}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "ac2ddaec-f1c9-4d29-962b-652561e2deff",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "Call:\n",
       "lm(formula = weight ~ height + I(height^2) + I(height^3), data = women)\n",
       "\n",
       "Residuals:\n",
       "     Min       1Q   Median       3Q      Max \n",
       "-0.40677 -0.17391  0.03091  0.12051  0.42191 \n",
       "\n",
       "Coefficients:\n",
       "              Estimate Std. Error t value Pr(>|t|)   \n",
       "(Intercept) -8.967e+02  2.946e+02  -3.044  0.01116 * \n",
       "height       4.641e+01  1.366e+01   3.399  0.00594 **\n",
       "I(height^2) -7.462e-01  2.105e-01  -3.544  0.00460 **\n",
       "I(height^3)  4.253e-03  1.079e-03   3.940  0.00231 **\n",
       "---\n",
       "Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1\n",
       "\n",
       "Residual standard error: 0.2583 on 11 degrees of freedom\n",
       "Multiple R-squared:  0.9998,\tAdjusted R-squared:  0.9997 \n",
       "F-statistic: 1.679e+04 on 3 and 11 DF,  p-value: < 2.2e-16\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fit3 <- lm(weight~height+I(height^2)+I(height^3),data=women)\n",
    "summary(fit3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "50fa75dd-24b2-45e8-b844-74fed68a20ea",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "Call:\n",
       "lm(formula = weight ~ height + height^2 + height^3, data = women)\n",
       "\n",
       "Residuals:\n",
       "    Min      1Q  Median      3Q     Max \n",
       "-1.7333 -1.1333 -0.3833  0.7417  3.1167 \n",
       "\n",
       "Coefficients:\n",
       "             Estimate Std. Error t value Pr(>|t|)    \n",
       "(Intercept) -87.51667    5.93694  -14.74 1.71e-09 ***\n",
       "height        3.45000    0.09114   37.85 1.09e-14 ***\n",
       "---\n",
       "Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1\n",
       "\n",
       "Residual standard error: 1.525 on 13 degrees of freedom\n",
       "Multiple R-squared:  0.991,\tAdjusted R-squared:  0.9903 \n",
       "F-statistic:  1433 on 1 and 13 DF,  p-value: 1.091e-14\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fit4 <- lm(weight~height+height^2+height^3,data=women)\n",
    "summary(fit4)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a55466ea-775c-4305-893b-9ed534aba51c",
   "metadata": {},
   "source": [
    "`lm(weight~height+height^2+height^3,data=women)`：R 语言会将height^2和height^3按照公式中的运算符号进行解析，把它们当作height的平方项和立方项与height一起作为自变量纳入模型。但这种方式可能会导致一些歧义或不符合预期的行为，因为 R 语言可能会尝试对height进行一些自动的转换或处理，在某些复杂情况下可能不是用户想要的结果。\n",
    "\n",
    "`lm(weight~height+I(height^2)+I(height^3),data=women)`：I()函数在这里的作用是告诉 R 语言将I()内部的表达式按照其字面意思进行解释，即直接将height的平方和立方作为独立的自变量，而不会进行额外的解析或转换。这样可以更明确地控制模型中自变量的形式，确保模型按照用户期望的方式构建。\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "96bc8079-b70a-472d-a31b-5a93b33a3a7c",
   "metadata": {},
   "outputs": [],
   "source": [
    "options(repos = c(CRAN = \"https://mirrors.aliyun.com/CRAN/\"))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "b4a84932-51a7-4094-9d24-b4c95818cd24",
   "metadata": {},
   "outputs": [],
   "source": [
    "# install.packages(\"car\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "102af8c3-dd5d-4689-a74e-c39d1947855b",
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Loading required package: carData\n",
      "\n"
     ]
    },
    {
     "data": {
      "image/png": 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ISQColzdERIYqW/jcfhyDnF945ULgya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",
      "text/plain": [
       "Plot with title \"women age 30-39\""
      ]
     },
     "metadata": {
      "image/png": {
       "height": 420,
       "width": 420
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "library(car)\n",
    "\n",
    "\n",
    "scatterplot(weight~height,data=women,\n",
    "            main=\"women age 30-39\",\n",
    "            xlab=\"height (inches)\",\n",
    "            ylab=\"weight(lbs)\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "23203cbb-1794-4c63-bfb9-535172d78015",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<style>\n",
       ".list-inline {list-style: none; margin:0; padding: 0}\n",
       ".list-inline>li {display: inline-block}\n",
       ".list-inline>li:not(:last-child)::after {content: \"\\00b7\"; padding: 0 .5ex}\n",
       "</style>\n",
       "<ol class=list-inline><li>'matrix'</li><li>'array'</li></ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 'matrix'\n",
       "\\item 'array'\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 'matrix'\n",
       "2. 'array'\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "[1] \"matrix\" \"array\" "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "class(state.x77)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "a62c646f-2436-4f81-b3c5-5555c1261ab9",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "'data.frame'"
      ],
      "text/latex": [
       "'data.frame'"
      ],
      "text/markdown": [
       "'data.frame'"
      ],
      "text/plain": [
       "[1] \"data.frame\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "class(as.data.frame(state.x77[,c(\"Murder\",\"Population\",\"Illiteracy\",\"Income\",\"Frost\")]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "bf5a8ace-9026-4f95-905f-af4318d54c2a",
   "metadata": {},
   "outputs": [],
   "source": [
    "states <- state.x77[,c(\"Murder\",\"Population\",\"Illiteracy\",\"Income\",\"Frost\")]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "f25509a3-5049-477a-9dbd-adadda7e96f2",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"dataframe\">\n",
       "<caption>A matrix: 5 × 5 of type dbl</caption>\n",
       "<thead>\n",
       "\t<tr><th></th><th scope=col>Murder</th><th scope=col>Population</th><th scope=col>Illiteracy</th><th scope=col>Income</th><th scope=col>Frost</th></tr>\n",
       "</thead>\n",
       "<tbody>\n",
       "\t<tr><th scope=row>Murder</th><td> 1.0000000</td><td> 0.3436428</td><td> 0.7029752</td><td>-0.2300776</td><td>-0.5388834</td></tr>\n",
       "\t<tr><th scope=row>Population</th><td> 0.3436428</td><td> 1.0000000</td><td> 0.1076224</td><td> 0.2082276</td><td>-0.3321525</td></tr>\n",
       "\t<tr><th scope=row>Illiteracy</th><td> 0.7029752</td><td> 0.1076224</td><td> 1.0000000</td><td>-0.4370752</td><td>-0.6719470</td></tr>\n",
       "\t<tr><th scope=row>Income</th><td>-0.2300776</td><td> 0.2082276</td><td>-0.4370752</td><td> 1.0000000</td><td> 0.2262822</td></tr>\n",
       "\t<tr><th scope=row>Frost</th><td>-0.5388834</td><td>-0.3321525</td><td>-0.6719470</td><td> 0.2262822</td><td> 1.0000000</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "A matrix: 5 × 5 of type dbl\n",
       "\\begin{tabular}{r|lllll}\n",
       "  & Murder & Population & Illiteracy & Income & Frost\\\\\n",
       "\\hline\n",
       "\tMurder &  1.0000000 &  0.3436428 &  0.7029752 & -0.2300776 & -0.5388834\\\\\n",
       "\tPopulation &  0.3436428 &  1.0000000 &  0.1076224 &  0.2082276 & -0.3321525\\\\\n",
       "\tIlliteracy &  0.7029752 &  0.1076224 &  1.0000000 & -0.4370752 & -0.6719470\\\\\n",
       "\tIncome & -0.2300776 &  0.2082276 & -0.4370752 &  1.0000000 &  0.2262822\\\\\n",
       "\tFrost & -0.5388834 & -0.3321525 & -0.6719470 &  0.2262822 &  1.0000000\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "A matrix: 5 × 5 of type dbl\n",
       "\n",
       "| <!--/--> | Murder | Population | Illiteracy | Income | Frost |\n",
       "|---|---|---|---|---|---|\n",
       "| Murder |  1.0000000 |  0.3436428 |  0.7029752 | -0.2300776 | -0.5388834 |\n",
       "| Population |  0.3436428 |  1.0000000 |  0.1076224 |  0.2082276 | -0.3321525 |\n",
       "| Illiteracy |  0.7029752 |  0.1076224 |  1.0000000 | -0.4370752 | -0.6719470 |\n",
       "| Income | -0.2300776 |  0.2082276 | -0.4370752 |  1.0000000 |  0.2262822 |\n",
       "| Frost | -0.5388834 | -0.3321525 | -0.6719470 |  0.2262822 |  1.0000000 |\n",
       "\n"
      ],
      "text/plain": [
       "           Murder     Population Illiteracy Income     Frost     \n",
       "Murder      1.0000000  0.3436428  0.7029752 -0.2300776 -0.5388834\n",
       "Population  0.3436428  1.0000000  0.1076224  0.2082276 -0.3321525\n",
       "Illiteracy  0.7029752  0.1076224  1.0000000 -0.4370752 -0.6719470\n",
       "Income     -0.2300776  0.2082276 -0.4370752  1.0000000  0.2262822\n",
       "Frost      -0.5388834 -0.3321525 -0.6719470  0.2262822  1.0000000"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "cor(states)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "10c713ce-5b2b-4d3f-9202-44ef755c2439",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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Ujzf07fqo6nIHWTgnThHsm+/PzxNr39+Fx3mubf79Pbd7cV/Ltt/f54BIX/Znfq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",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 420,
       "width": 420
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "scatterplotMatrix(states)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "3257fc76-8c21-4712-8c4b-3e021cc1ce79",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "Call:\n",
       "lm(formula = Murder ~ Population + Illiteracy + Income + Frost, \n",
       "    data = states)\n",
       "\n",
       "Residuals:\n",
       "    Min      1Q  Median      3Q     Max \n",
       "-4.7960 -1.6495 -0.0811  1.4815  7.6210 \n",
       "\n",
       "Coefficients:\n",
       "             Estimate Std. Error t value Pr(>|t|)    \n",
       "(Intercept) 1.235e+00  3.866e+00   0.319   0.7510    \n",
       "Population  2.237e-04  9.052e-05   2.471   0.0173 *  \n",
       "Illiteracy  4.143e+00  8.744e-01   4.738 2.19e-05 ***\n",
       "Income      6.442e-05  6.837e-04   0.094   0.9253    \n",
       "Frost       5.813e-04  1.005e-02   0.058   0.9541    \n",
       "---\n",
       "Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1\n",
       "\n",
       "Residual standard error: 2.535 on 45 degrees of freedom\n",
       "Multiple R-squared:  0.567,\tAdjusted R-squared:  0.5285 \n",
       "F-statistic: 14.73 on 4 and 45 DF,  p-value: 9.133e-08\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "states <- as.data.frame(states)\n",
    "fit <- lm(Murder~Population+Illiteracy+Income+Frost,data=states)\n",
    "summary(fit)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5f527c10-7e10-4d52-8c49-c1977e828d70",
   "metadata": {},
   "source": [
    "$t_i = \\frac{\\hat\\beta_i}{s_{\\hat\\beta_i}}$,这里的Std. Error 实际上是$s_{\\hat\\beta_i}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "ced5b320-1e4e-423a-bdd4-2013833917aa",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "2.4712770658418"
      ],
      "text/latex": [
       "2.4712770658418"
      ],
      "text/markdown": [
       "2.4712770658418"
      ],
      "text/plain": [
       "[1] 2.471277"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "2.237e-04/9.052e-05"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "b9149de7-e025-4827-9948-6a4d568d1aa6",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "4.73810612991766"
      ],
      "text/latex": [
       "4.73810612991766"
      ],
      "text/markdown": [
       "4.73810612991766"
      ],
      "text/plain": [
       "[1] 4.738106"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "4.143e+00/8.744e-01 "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "74482335-85c7-4562-b5e2-e52c0c15a3b0",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "Call:\n",
       "lm(formula = mpg ~ hp + wt + hp:wt, data = mtcars)\n",
       "\n",
       "Residuals:\n",
       "    Min      1Q  Median      3Q     Max \n",
       "-3.0632 -1.6491 -0.7362  1.4211  4.5513 \n",
       "\n",
       "Coefficients:\n",
       "            Estimate Std. Error t value Pr(>|t|)    \n",
       "(Intercept) 49.80842    3.60516  13.816 5.01e-14 ***\n",
       "hp          -0.12010    0.02470  -4.863 4.04e-05 ***\n",
       "wt          -8.21662    1.26971  -6.471 5.20e-07 ***\n",
       "hp:wt        0.02785    0.00742   3.753 0.000811 ***\n",
       "---\n",
       "Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1\n",
       "\n",
       "Residual standard error: 2.153 on 28 degrees of freedom\n",
       "Multiple R-squared:  0.8848,\tAdjusted R-squared:  0.8724 \n",
       "F-statistic: 71.66 on 3 and 28 DF,  p-value: 2.981e-13\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fit <- lm(mpg~hp+wt+hp:wt,data=mtcars)\n",
    "summary(fit)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "85a8f247-4cb5-4356-893e-2e1f78659679",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "Call:\n",
       "lm(formula = mpg ~ (hp + wt)^2, data = mtcars)\n",
       "\n",
       "Residuals:\n",
       "    Min      1Q  Median      3Q     Max \n",
       "-3.0632 -1.6491 -0.7362  1.4211  4.5513 \n",
       "\n",
       "Coefficients:\n",
       "            Estimate Std. Error t value Pr(>|t|)    \n",
       "(Intercept) 49.80842    3.60516  13.816 5.01e-14 ***\n",
       "hp          -0.12010    0.02470  -4.863 4.04e-05 ***\n",
       "wt          -8.21662    1.26971  -6.471 5.20e-07 ***\n",
       "hp:wt        0.02785    0.00742   3.753 0.000811 ***\n",
       "---\n",
       "Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1\n",
       "\n",
       "Residual standard error: 2.153 on 28 degrees of freedom\n",
       "Multiple R-squared:  0.8848,\tAdjusted R-squared:  0.8724 \n",
       "F-statistic: 71.66 on 3 and 28 DF,  p-value: 2.981e-13\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "summary(lm(mpg~(hp+wt)^2,data=mtcars))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "8f5e2cd3-8248-49b1-81d4-d97f125b6741",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "32"
      ],
      "text/latex": [
       "32"
      ],
      "text/markdown": [
       "32"
      ],
      "text/plain": [
       "[1] 32"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "nrow(mtcars)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "06996058-a539-4ae0-9531-51407b8624a5",
   "metadata": {},
   "outputs": [],
   "source": [
    "# install.packages(\"effects\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "id": "0db2cabb-cc38-4050-8170-661742662ab6",
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "lattice theme set by effectsTheme()\n",
      "See ?effectsTheme for details.\n",
      "\n"
     ]
    },
    {
     "data": {
      "image/png": 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cWSR0j5pERJkeQSUpnLDI8DpTgyCimT\nlCgpiqxCymSGR0kRZBZSHilRUnjZhVTmMMOjpOAEQ3JP20PANVog+ZQ4UArNT0jOFeHWaJHU\nZ3iUFJjk1G5fnG9/ngv3W+7c4n1SmJCySCn2KuREMKSD+6v//nPb8uo2wdZoubRToqSQRKd2\nnQtucQ8BQ0o8JUoKSDCk4rFHKsyElHZKHCiFIzq1ux8jHcqf2/Qu1BqtlfDBEiUFI3myYXs/\n+V3tkE7B1mi9tFOKvQp5EH1C9ry7ZbSrdkvuGG6NRCSbEiWFkeMrG4almhIlBUFIT4nO8DhQ\nCkE0pJ9qarc/r1idLwP4lmZKlBSAj5MNuzUrFDWkMtEZHiV55+X09+IzdovWSFqKKVGSb16e\nkF388qDPAwST4AyPkjzz9RKh5RSElGJKHCj55eVFq6sOklSEVKY3w6MkryRPNhzrY6TfYvGr\ng74NEFiCKcVehXT5+sW+oendaeOKw7W+eCgeF1evkUeJzfAoyZ+AIR2aX52t8mlOlA+fk9AU\nUmopUZI34V7Z8Of2t4ZObn+b/bnir/yrfpNWcgBfUkqJAyVfwoW0a25Z7aoOrjqY+hl+Zau6\nkJJKiZI8Cf5auyqknbuUo2f3FIaU1AyPkryQDOlQjJ9ouLs2v63ULGrw36kMKaWUKMkH0eeR\nPpyxuztVs7qBkD6e7lMilZQoyYPZD+BPZ+2+v8buUuxKm3ukWiIpcaAkz8dLhMZdmydrzYaU\nygyPksSJTu2Gn2Ht2DZPHRV2Q0oopdirkBbR30faXj7e9LJp/0Fz1u5i6KxdXwotUZIsyZDO\nnw+3zo+36DrWzyOdh9/W2EBISaRESaIEQzp+Pm9xeb7Vna1XNowwnxIHSpJEf7Hv41m7faey\nzf397yTWKBrruyVKEhTurF13d3WtX/0ttEYRJZBS7FVIhejU7utZu3UDqGS7JUqSIvqLfdvB\ngx65AXQynRIlCfHz+0hB10gBwylxoCSDkGTY3S1RkgjesliK6ZRir4J9hCTIakuUtB4hiTKa\nEtO71QhJmsmUKGktQpJncrdESesQkg8WU6KkVQjJE3stUdIahOSNuZQ4UFqBkHwylhIlLUdI\nftlLKfYqGEVIvtma4VHSQoTkn6mUKGkZQgrCUEocKC1CSIHYSYmSliCkYOzM8ChpPkIKyExK\nlDQbIYVlJCVKmouQQrOREgdKMxFSeCZmeJQ0DyHFYCWl2KtgCCFFYiAlSpqBkKLRnxIlTUdI\nEamf4XGgNBkhRaU9JUqaipBi059S7FUwgZDi050SJU1CSBqonuFR0hSEpIPmlChpAkJSQ29K\nnHL4jpAUUZsSJX1FSKpoTin2KuhGSMpoPViipM8ISR2lKVHSR4SkkcqUKOkTQtJJY0qccviA\nkLRSOMOjpHGEpJfOlGKvglKEpJq6lChpBCEppy0lShpGSOopm+FR0iBCMkBXSpxyGEJINmhK\niZIGEJIVylKKvQraEJIdimZ4lPSKkCzRkxIlvSAkY7SkxIFSHyGZoyQlSuohJIOUzPAoqYOQ\nTNKREiU9EZJVGlKipAdCsktBSpR0R0iWxZ/hccqhRUi2RU+JkhqEZJ6ClCKOrgUhJSBySpRU\nElIi4s7wKImQkhE1JUoipIRETIlTDoSUkni7pexLIqS0RE0pxrBaEFJyYqWUd0mElKBIKWVd\nEiElKc4ML+cDJUJKVJSUMi6JkNIVKaXAIypBSCmLsFvKtSRCSlv4lDItiZCSFzqlPA+UCCkD\ngVPKsiRCykLgGV6GJRFSJsKmlF9JhJSPkCllVxIh5SRgSrkdKBFSXsLN8DIriZByEzSlEMPo\nQEgZCpVSTiURUpYCpZRRSYSUqTAzvHwOlAgpW0FSyqYkQspZoJQ8j6ACIeUtQEp5lERIufM/\nw8tiekdI8J5SDiUREkr/M7z0SyIk1DynlHxJhISW3xle6iUREh68ppT4gRIhoctjSmmXREjo\n85uSnwUrQEh45W+Gl3BJhIR33lJKtyRCwiBPKSV7oERIGOEnpVRLIiSM8jPDS7MkQsIHXlJK\nsiRCwmceUkqxJELCN/K7pQQPlAgJ34mnlF5JhIRJPKQkuLT4CAkTCe+WEiuJkDCZbEpplURI\nmEMypaQOlAgJ8wjullIqiZAwl2xKIsuJj5CwgFhLyZRESFhEKqVUSiIkLCWTUiIHSoSE5UR2\nS2mUREhYQyolkZWJiZCwkkBKCZRESFht/W7JfkmEBAGrUzJ/oERIkLEyJeslERKkrE9Jbl2C\nIyTIWTfDM10SIUHSqpQsl0RIELYiJcMHSoQEcctTslsSIcGD5TM8qyURErxYnJLRkggJvixM\nyWZJhAR/lqVk8kCJkODTohmexZIICX4tTcnLyvhDSPBuQUrmSiIkBDA/JWslERKCmD3DM3ag\nREgIZG5KtkoiJIQzPyV/6yKMkBDSvJQMlURICGvWDM9OSYSE0OakZOZAiZAQwfSUrJRESIhi\nVkqe10UCISGSyTM8EyUREqKZmpKFkggJMU1LycCBEiEhrkkp6S+JkBDbpBme9pIICfFNSUl5\nSYQEFb6npLskQoISX3dLqg+UCAlqfEtJc0mEBE2+pxRuXWYJG9KpvbVryA8A8z6npLakoCH9\nte38ERLGfZzhaS0pZEh/xSOk3fQB/jdm+YpAuU8pKT1QChjSyW3bkE7uOH2A0ZAILGXjKeks\nKWBI7lA+QjqtH4C+Eje+W9JYUsCQ/sp7SDt33rviID1Ag8CSMZqSwpLCnrV7hFTb9q55WDPA\nOPoyaSQlfSXNfgBLhOTcT1leD8MTvMBn7QhMueGU1B0oRdkjNa5uIz6AmLl9EZg/gzM8bSVF\nDKkc3gnqCGkMgcUwllKUlRlGSDIIy7OBlFSVFCWkwl1vf16Gn5a1GdIYuhLznpKmkqKEdHCH\n+mTDWXwA9QhrhbcZnqIDpSghXYv6NOHwE0lph/SCCeE8rynpKSnOMdL1ULjNyKsbsgrp1dSw\nMg7sPaV469LB7yNZMLevtAPrp6SkJEKyLNfAejM8HdM7QkpR+n11U1JREiHlJKnA+inFXRdC\nQmm3r2dK8UsiJIzSH9hjhhe9JELCXKr6uqcU+0CJkCAlVmBNSpFLIiR4NrevBYE9UpJf+6kI\nCZGIBtbM8CKWREhQZmFfbUqx1pqQYITkDkweIcE2JX0REpLyeGYpcGCEhMR8/nAYX2ERElLz\n79+0z0rvWRsWISE5/xal1Dc3LEJCev5JpNRHSMhR9SqHKZ+VLoaQkKT69UIBUyIkpKl95V2o\nlAgJifoXNCVCQqoev1gRYoZHSEjX4xcr/KdESEhY51eUPKdESEhZ95f9vKZESEha79dmPc7w\nCAlp+xcmJUJC4l7fFcVPSoSE5L2+K4qPlAgJ6Xt7fyH5GR4hIQPv79QlnRIhIQdD73knmhIh\nIQuDb8QqmBIhIQ/Db2ksNsMjJORi+C2NhVIiJGRj7M3BJVIiJORj9G3216dESMjI+AdWrJ3h\nERJy8uFTlNalREjIysfPI1vREiEhMx8/j2xxSoSE3Hz5ZL9lKRESsvPtMzKX7JYICfn5+sHN\n81MiJGRowkegz2yJkJClCR+BPislQkKeJpQ058QDISFTk0qavFsiJORqWkkTUyIkZGvCKYfG\nhJYICfmaXNL3lAgJOZtc0rcTD4SErM0o6eNuiZCQtzklfUiJkJC5WSWNtkRIyN30Uw6NwZQI\nCdmbW1Ld0st3CAmYO70bQEiAQEmEBJTrSyIkoDL/QKmHkIDaupIICWitKYmQgLsVJRES8LB8\nekdIwNPikggJ6FpYEiEBPctKIiSgb9H0jpCAF0tKIiTgzfySCAl4N7skQgIGzC2JkIAhMw+U\nCAkYNK8kQgJGzCmJkIAxM0oiJGDU9OkdIQHjJpdESMAnE0siJOCjaSUREvDZpOkdIQFfTCmJ\nkICvvpdESMB3X0siJGCCbyUREjDFlwMlQgIm+VwSIQETfSqJkICpPpRESMBk49M7QgKmGy2J\nkIA5RkoiJGCW4ZIICZhncHpHSMBMQyUREjDbe0mEBMxHSIAP+kMaG9C1f7rFa+TufzwujI3s\n7oMMj7V8Fdbd2o1cXryQWcubMOKcu2bgf6CfVpGQSkKa8e15IxKS4A3WDkBI47ccubx4IbOW\nR0hTr5O5wdoBCGn8liOXFy9k1vIIaep1MjdYOwAhjd9y5PLihcxaHiFNvU7mBmsHIKTxW45c\nXryQWcsjpKnXydxg7QCENH7LkcuLFzJreYQ09TqZG6wdgJDGbzlyefFCZi2PkKZeJ3ODtQMQ\n0vgtRy4vXsis5RHS1OtkbrB2AEIav+XI5cULmbU8Qpp6ncwN1g5ASOO3HLm8eCGzlkdIU6+T\nucHaAQhp/JYjlxcvZNbyCGnqdTI3mD0AYNDsx7mPeFQMxpAMGRAhMSRDCiAkhmRIAYTEkAwp\ngJAYkiEFEBJDMqQAQmJIhhRASAzJkAIIiSEZUgAhMSRDClC9coAVhAQIICRAACEBAggJEEBI\ngABCAgQQEiCAkAABhAQICBRS9x0lDoUrDlePg53ud6ozkudB70MGu5+nzdBdCzRkqHt53Tu3\n/ytfx/H+AFoiTEh/nR/9tr608TpYc6EzkudB70MGu5+HeuHFtQx3L59DBruXRb3wv5dxvD+A\nFgkV0u5+8dcVf+Vf4X69jVW0j+rOSJ4HfQwZ6n7+uf212g3uw93LzpCh7uWhGuxQDxZuWy4V\nJqSTO94vHtz59ufP8xviQ23vk47nSH4HfQ4Z6n7u2rfTdOHuZWfIUPeycNd2xHDbcrFQIZ3u\nF3fuUvb+pybMHe5vCNoZye+gzyED3s96YBfwXj6HDHsvXVEGv5cLhAlp58772wFiPeDz/2xe\n/L0OUf3ld9DnkAHv583VbQPey+eQQe/loa427L1cIlRItW0Z5OcQOqSyE1LI+3mqJjlhH2L1\nkAHv5Y9zr8FmHZJzP7f/m9X/c0k6pKD381LsysAPsfuQwe7laVfUB0OE1HWtzlkmHVIjzP28\nFtvOAEHuZTtk+0Wgrbl/CZaQmjtfBAupM5L3QftLDjLktnkmJeS93PaevAm0Na/V2YaQ93KZ\n4CE1J10uXs/zdM/aXZ5nejwO+h6S5yEvm+2lvhDuXj6GbAXamq/jhBhygTAhNc8I1Hf+WD8N\ncG6OIf1oH9WdkbwP+tgJBrqfZ3efZAW7l88hQ93L+zibsNtymTAhHaq7fa2fSgvwxHTwVzY8\nhgx1Py+PB3Wwe9kZMtS9rF/ZcN1Vx0i8sqFxbV41Vf9PZPM4d+rLfZ7VGcn3oO2Qoe7nvvO5\ncoHuZWfIYFuzGLxr/h9ASwQ6RroeCrc5PS4WXvfL95A6I/ketDtkgPvpno/qUPfydcggW3Nw\nHP8PoCWUnfsAbCIkQAAhAQIICRBASIAAQgIEEBIggJAAAYQECCAkQAAhAQIICRBASIAAQgIE\nEBIggJAAAYQECCAkQAAhAQIICRBASOZoe49RVNgo5hCSRmwUcwhJIzaKOYSkERvFnFtIB1cc\nH5fUvVdilgjJHOfqj8yrPzXoqPHde7NESObcyrmWp+Zjvpr3k/+JvUogJHtc/UEMzYfXNZ9w\nouyjgrJESOY0JxsUfwpkltgG5hCSRmwDcwhJI7aBOd2QqqOlc/XBdoiMkMzphtSctTvHXiUQ\nkj3dkLbV80ictFOAkMzpHSPt7p8NibgIyTDOMujBpjCMkPRgUxhGSHqwKQwjJD3YFIAAQgIE\nEBIggJAAAYQECCAkQAAhAQIICRBASIAAQgIEEBIggJAAAYQECCAkQAAhAQIICRBASIAAQgIE\nEBIggJAAAYQECCAkQAAhAQIICRBASIAAQgIE/B/tODaIUw6jIQAAAABJRU5ErkJggg==",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 420,
       "width": 420
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "library(effects)\n",
    "plot(effect(\"hp:wt\",fit,,list(wt=c(2.2,3.2,4.2))),multiline=TRUE)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "id": "1a435a0c-36f8-4d0b-9a36-a1ac05dbc0fd",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"dataframe\">\n",
       "<caption>A matrix: 4 × 2 of type dbl</caption>\n",
       "<thead>\n",
       "\t<tr><th></th><th scope=col>2.5 %</th><th scope=col>97.5 %</th></tr>\n",
       "</thead>\n",
       "<tbody>\n",
       "\t<tr><th scope=row>(Intercept)</th><td> 42.42359654</td><td>57.19325031</td></tr>\n",
       "\t<tr><th scope=row>hp</th><td> -0.17069436</td><td>-0.06950982</td></tr>\n",
       "\t<tr><th scope=row>wt</th><td>-10.81750352</td><td>-5.61574508</td></tr>\n",
       "\t<tr><th scope=row>hp:wt</th><td>  0.01264983</td><td> 0.04304647</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "A matrix: 4 × 2 of type dbl\n",
       "\\begin{tabular}{r|ll}\n",
       "  & 2.5 \\% & 97.5 \\%\\\\\n",
       "\\hline\n",
       "\t(Intercept) &  42.42359654 & 57.19325031\\\\\n",
       "\thp &  -0.17069436 & -0.06950982\\\\\n",
       "\twt & -10.81750352 & -5.61574508\\\\\n",
       "\thp:wt &   0.01264983 &  0.04304647\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "A matrix: 4 × 2 of type dbl\n",
       "\n",
       "| <!--/--> | 2.5 % | 97.5 % |\n",
       "|---|---|---|\n",
       "| (Intercept) |  42.42359654 | 57.19325031 |\n",
       "| hp |  -0.17069436 | -0.06950982 |\n",
       "| wt | -10.81750352 | -5.61574508 |\n",
       "| hp:wt |   0.01264983 |  0.04304647 |\n",
       "\n"
      ],
      "text/plain": [
       "            2.5 %        97.5 %     \n",
       "(Intercept)  42.42359654 57.19325031\n",
       "hp           -0.17069436 -0.06950982\n",
       "wt          -10.81750352 -5.61574508\n",
       "hp:wt         0.01264983  0.04304647"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "confint(fit)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "id": "eca8c572-4cda-4ebd-84a5-21e65623fe4a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"dataframe\">\n",
       "<caption>A matrix: 3 × 2 of type dbl</caption>\n",
       "<thead>\n",
       "\t<tr><th></th><th scope=col>2.5 %</th><th scope=col>97.5 %</th></tr>\n",
       "</thead>\n",
       "<tbody>\n",
       "\t<tr><th scope=row>(Intercept)</th><td>33.95738245</td><td>40.49715778</td></tr>\n",
       "\t<tr><th scope=row>hp</th><td>-0.05024078</td><td>-0.01330512</td></tr>\n",
       "\t<tr><th scope=row>wt</th><td>-5.17191604</td><td>-2.58374544</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "A matrix: 3 × 2 of type dbl\n",
       "\\begin{tabular}{r|ll}\n",
       "  & 2.5 \\% & 97.5 \\%\\\\\n",
       "\\hline\n",
       "\t(Intercept) & 33.95738245 & 40.49715778\\\\\n",
       "\thp & -0.05024078 & -0.01330512\\\\\n",
       "\twt & -5.17191604 & -2.58374544\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "A matrix: 3 × 2 of type dbl\n",
       "\n",
       "| <!--/--> | 2.5 % | 97.5 % |\n",
       "|---|---|---|\n",
       "| (Intercept) | 33.95738245 | 40.49715778 |\n",
       "| hp | -0.05024078 | -0.01330512 |\n",
       "| wt | -5.17191604 | -2.58374544 |\n",
       "\n"
      ],
      "text/plain": [
       "            2.5 %       97.5 %     \n",
       "(Intercept) 33.95738245 40.49715778\n",
       "hp          -0.05024078 -0.01330512\n",
       "wt          -5.17191604 -2.58374544"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "confint(lm(mpg~hp+wt,data=mtcars))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "id": "6465f3e1-aefe-4672-9cff-61a49cc2bfe2",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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iTZY4gXqfpJq1YhJNjoiDkQ7y2LEan5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\nAIgEIAAiAQiASAAC/B9dD0r4JvQbuwAAAABJRU5ErkJggg==",
      "text/plain": [
       "Plot with title \"\""
      ]
     },
     "metadata": {
      "image/png": {
       "height": 420,
       "width": 420
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fit<-lm(weight~height,data=women)\n",
    "par(mfrow=c(2,2))\n",
    "plot(fit)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "id": "4f639eab-8d3c-4f8e-ac18-3b6e88e2588c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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TcltkjmsSxI3WEFiDR8IxwlPRRuHrXT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\nAAAAAElFTkSuQmCC",
      "text/plain": [
       "Plot with title \"\""
      ]
     },
     "metadata": {
      "image/png": {
       "height": 420,
       "width": 420
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fit2<-lm(weight~height+I(height^2),data=women)\n",
    "par(mfrow=c(2,2))\n",
    "plot(fit2)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "18e29e4d-73ab-4db7-aeb6-5d7337602f41",
   "metadata": {},
   "source": [
    "上面四幅图分别检查线性回归的假设：线性、正态性、同方差性\n",
    "\n",
    "- 残差与拟合值图：检查线性，拟合良好的模型，其残差点应均匀分布在这条虚线的两侧；\n",
    "- 正态 Q-Q 图：检查正态性,图中横坐标是理论上服从正态分布的数据的分位数，纵坐标是实际观测到的残差的分位数。；\n",
    "- 尺度 - 位置图:检查同方差性，一般是一条平滑的曲线，用于拟合数据点的整体趋势，帮助观察数据的分布模式。如果数据满足同方差性，这条曲线应该是大致水平的；\n",
    "- 残差与杠杆值图：检验离群点、杠杆值和强影响点。\n",
    "  \n",
    "独立性这里没有体现检验的过程，可以按照采样的过程进行判断\n",
    "\n",
    "线性回归是一种基本的统计分析方法，有以下主要假设：\n",
    "\n",
    "#### 线性关系假设\n",
    "- **自变量与因变量呈线性关系**：假设因变量$y$与自变量$x_1,x_2,\\cdots,x_p$之间存在线性关系，可表示为$y=\\beta_0+\\beta_1x_1+\\beta_2x_2+\\cdots+\\beta_px_p+\\epsilon$，其中$\\beta_0,\\beta_1,\\cdots,\\beta_p$是回归系数，$\\epsilon$是误差项。\n",
    "- **实际意义**：意味着自变量的变化对因变量的影响是均匀的、成比例的。如在研究身高与体重的关系时，假设体重与身高呈线性关系，即身高每增加一定单位，体重会按照固定的比例增加。\n",
    "\n",
    "#### 独立性假设\n",
    "- **观测值相互独立**：要求每个观测值$(x_{i1},x_{i2},\\cdots,x_{ip},y_i)$之间相互独立，即一个观测值的取值不会影响其他观测值的取值。在时间序列数据中，可能会存在自相关性，即当前观测值与过去的观测值相关，此时就不满足独立性假设。\n",
    "- **实际意义**：保证样本数据是随机抽样得到的，不存在系统性的关联，使模型的估计和推断具有可靠性。\n",
    "\n",
    "#### 正态性假设\n",
    "- **误差项服从正态分布**：假定误差项$\\epsilon$服从均值为$0$，方差为$\\sigma^2$的正态分布，即$\\epsilon\\sim N(0,\\sigma^2)$。这意味着大部分误差项的值集中在均值$0$附近，离均值越远，出现的概率越小。\n",
    "- **实际意义**：基于正态分布的性质，可以进行参数的假设检验和置信区间估计等统计推断，为模型的有效性和可靠性提供理论依据。\n",
    "\n",
    "#### 等方差性假设\n",
    "- **误差项方差恒定**：要求在不同的自变量取值下，误差项$\\epsilon$的方差始终保持不变，即$Var$epsilon_i)=\\sigma^2$，$i = 1,2,\\cdots,n$。如果误差项的方差随着自变量的变化而变化，就会出现异方差问题。\n",
    "- **实际意义**：保证回归模型的估计量具有最小方差等良好的统计性质，使模型的预测和推断更加准确可靠。\n",
    "\n",
    "#### 无多重共线性假设\n",
    "- **自变量间不存在高度线性相关**：假设自变量$x_1,x_2,\\cdots,x_p$之间不存在严格的线性关系，即不存在一个自变量可以由其他自变量线性表示的情况。若存在多重共线性，会导致回归系数的估计不准确，标准误差增大，模型的稳定性和解释力下降。\n",
    "- **实际意义**：确保每个自变量对因变量的影响能够被独立地估计和解释，使模型具有良好的可解释性和预测能力。\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "id": "a3b982a6-660f-4124-9c22-a095454860ce",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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tyj57nNlsUHi9xrs7bbfPJed3CTiPru0nqxiEAmIlV/fz6uLi+/2z/q4Z0mWX7W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\ns++St/QAAAAASUVORK5CYII=",
      "text/plain": [
       "Plot with title \"\""
      ]
     },
     "metadata": {
      "image/png": {
       "height": 420,
       "width": 420
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fit3<-lm(weight~height,data=women[-c(2,13,15),])\n",
    "par(mfrow=c(2,2))\n",
    "plot(fit3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "id": "1f5c52ce-c95a-43b5-b8db-720a41792617",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "Call:\n",
       "lm(formula = weight ~ height, data = women[-c(2, 13, 15), ])\n",
       "\n",
       "Residuals:\n",
       "    Min      1Q  Median      3Q     Max \n",
       "-1.3373 -0.7580 -0.3373  0.3154  2.6148 \n",
       "\n",
       "Coefficients:\n",
       "             Estimate Std. Error t value Pr(>|t|)    \n",
       "(Intercept) -80.84830    6.36560  -12.70 1.71e-07 ***\n",
       "height        3.34132    0.09853   33.91 1.18e-11 ***\n",
       "---\n",
       "Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1\n",
       "\n",
       "Residual standard error: 1.273 on 10 degrees of freedom\n",
       "Multiple R-squared:  0.9914,\tAdjusted R-squared:  0.9905 \n",
       "F-statistic:  1150 on 1 and 10 DF,  p-value: 1.176e-11\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "summary(fit3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "id": "9f46bfbd-7eb8-49db-a007-15efa5122d6d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "Call:\n",
       "lm(formula = Murder ~ Population + Illiteracy + Income + Frost, \n",
       "    data = states)\n",
       "\n",
       "Residuals:\n",
       "    Min      1Q  Median      3Q     Max \n",
       "-4.7960 -1.6495 -0.0811  1.4815  7.6210 \n",
       "\n",
       "Coefficients:\n",
       "             Estimate Std. Error t value Pr(>|t|)    \n",
       "(Intercept) 1.235e+00  3.866e+00   0.319   0.7510    \n",
       "Population  2.237e-04  9.052e-05   2.471   0.0173 *  \n",
       "Illiteracy  4.143e+00  8.744e-01   4.738 2.19e-05 ***\n",
       "Income      6.442e-05  6.837e-04   0.094   0.9253    \n",
       "Frost       5.813e-04  1.005e-02   0.058   0.9541    \n",
       "---\n",
       "Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1\n",
       "\n",
       "Residual standard error: 2.535 on 45 degrees of freedom\n",
       "Multiple R-squared:  0.567,\tAdjusted R-squared:  0.5285 \n",
       "F-statistic: 14.73 on 4 and 45 DF,  p-value: 9.133e-08\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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GVe/nXdvxfzvrFCtkiaKq0FY/m8m5f4lRMWMKzdFiUiGdM3RT+pNQSRuJjKJ6qc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",
      "text/plain": [
       "Plot with title \"\""
      ]
     },
     "metadata": {
      "image/png": {
       "height": 420,
       "width": 420
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fit <- lm(Murder~Population+Illiteracy+Income+Frost,data=states)\n",
    "summary(fit)\n",
    "par(mfrow=c(2,2))\n",
    "plot(fit)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "id": "b41bc672-a540-4396-8754-b279ae0454cc",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "Call:\n",
       "lm(formula = Murder ~ Population + Illiteracy + Income + Frost, \n",
       "    data = states[current_rows, ])\n",
       "\n",
       "Residuals:\n",
       "    Min      1Q  Median      3Q     Max \n",
       "-4.3849 -1.6753 -0.1691  1.5937  4.4115 \n",
       "\n",
       "Coefficients:\n",
       "              Estimate Std. Error t value Pr(>|t|)    \n",
       "(Intercept)  3.037e+00  3.486e+00   0.871  0.38838    \n",
       "Population   2.481e-04  8.104e-05   3.061  0.00375 ** \n",
       "Illiteracy   4.020e+00  7.807e-01   5.150 5.87e-06 ***\n",
       "Income      -2.599e-04  6.167e-04  -0.421  0.67546    \n",
       "Frost       -4.133e-03  9.066e-03  -0.456  0.65071    \n",
       "---\n",
       "Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1\n",
       "\n",
       "Residual standard error: 2.261 on 44 degrees of freedom\n",
       "Multiple R-squared:  0.6541,\tAdjusted R-squared:  0.6226 \n",
       "F-statistic:  20.8 on 4 and 44 DF,  p-value: 1.108e-09\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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RAJAAIgEgAEQCQACIBIABAAkQAgACIBQABEAoAAiAQAARAJAAIgEgAEQCQACIBI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\nCIBIABAAkQAgACIBQMD/AZnsVLkZVC3JAAAAAElFTkSuQmCC",
      "text/plain": [
       "Plot with title \"\""
      ]
     },
     "metadata": {
      "image/png": {
       "height": 420,
       "width": 420
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "current_rows <- !(rownames(states) %in% c(\"Nevada\"))\n",
    "fit <- lm(Murder~Population+Illiteracy+Income+Frost,data=states[current_rows,])\n",
    "summary(fit)\n",
    "par(mfrow=c(2,2))\n",
    "plot(fit)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "392056bf-840c-4ca7-9e3d-ff9bd5093098",
   "metadata": {},
   "source": [
    "### Durbin - Watson检验\n",
    "\n",
    "简称D - W检验，是一种用于检验回归模型中误差项是否存在自相关的方法，以下是其详细介绍：\n",
    "\n",
    "#### 基本原理\n",
    "- 该检验基于残差序列进行，其基本思想是通过分析回归模型残差之间的相关性来判断误差项是否存在自相关。如果误差项不存在自相关，那么相邻残差之间应该是相互独立的，不存在明显的规律；反之，如果存在自相关，残差之间就会呈现出一定的相关性。\n",
    "- 其统计量$d$的计算公式为：$d=\\frac{\\sum_{t = 2}^{n}(e_{t}-e_{t - 1})^{2}}{\\sum_{t = 1}^{n}e_{t}^{2}}$，其中$e_t$是第$t$期的残差，$n$是样本容量。$d$统计量的取值范围在$0$到$4$之间。\n",
    "\n",
    "#### 检验规则\n",
    "- **正自相关判断**：当$d$值接近$0$时，表明存在正自相关，即相邻残差倾向于具有相同的符号，误差项呈现出正的相关性。\n",
    "- **无自相关判断**：当$d$值接近$2$时，表明不存在自相关，此时残差之间相互独立，符合回归模型的基本假设。\n",
    "- **负自相关判断**：当$d$值接近$4$时，表明存在负自相关，即相邻残差倾向于具有相反的符号，误差项呈现出负的相关性。\n",
    "- **不确定区域**：在$d$值的某些区间内，无法明确判断是否存在自相关，需要进一步的分析或采用其他检验方法辅助判断。具体的临界值会根据样本容量$n$和自变量的个数$k$等因素而有所不同，可通过查阅Durbin - Watson检验临界值表来确定。\n",
    "\n",
    "#### 应用场景\n",
    "- **经济数据分析**：在经济时间序列分析中，如分析国内生产总值（GDP）、通货膨胀率、失业率等经济指标的时间序列数据时，Durbin - Watson检验可帮助判断模型是否存在自相关问题，以确保模型的有效性和可靠性。\n",
    "- **金融风险预测**：在金融领域，对股票价格、汇率、利率等金融数据进行建模分析时，该检验可用于检验模型残差的自相关性，有助于准确评估金融风险，提高预测的准确性。\n",
    "- **工程与科学研究**：在工程领域的质量控制、信号处理以及自然科学研究中的实验数据建模等方面，Durbin - Watson检验也被广泛应用，用于检验模型的合理性和数据的可靠性。\n",
    "\n",
    "#### 局限性\n",
    "- **对高阶自相关检验效果有限**：Durbin - Watson检验主要适用于检验一阶自相关，对于高阶自相关的检验效果可能不理想。如果存在高阶自相关，可能需要采用其他更复杂的检验方法，如Ljung - Box检验等。\n",
    "- **依赖模型设定**：该检验的有效性依赖于回归模型的正确设定，如果模型存在遗漏变量、函数形式错误等问题，可能会导致检验结果不准确。\n",
    "- **样本容量要求**：在样本容量较小时，Durbin - Watson统计量的分布可能会偏离其理论分布，从而影响检验的可靠性。\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "id": "40432e7b-6bec-46f3-9cdb-f2a985132c20",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       " lag Autocorrelation D-W Statistic p-value\n",
       "   1      -0.1598535      2.210053    0.42\n",
       " Alternative hypothesis: rho != 0"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "durbinWatsonTest(fit)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "148ee2cf-9104-43c1-a4ca-350e17425245",
   "metadata": {},
   "source": [
    "在Durbin - Watson检验相关的返回结果中，`lag`和`Autocorrelation`具有如下含义：\n",
    "\n",
    "#### lag（滞后阶数）\n",
    "- **定义**：指在时间序列分析中，当前观测值与之前某一时期观测值之间的时间间隔。在自相关分析的情境下，它表示计算自相关时所考虑的时间滞后长度。\n",
    "- **作用**：通过设定不同的`lag`值，可以观察时间序列在不同滞后阶数下的自相关情况，帮助分析数据中的潜在模式和规律。例如，`lag = 1`表示考虑当前观测值与前一期观测值之间的相关性；`lag = 2`表示考虑当前观测值与前两期观测值之间的相关性，以此类推。\n",
    "\n",
    "#### Autocorrelation（自相关）\n",
    "- **定义**：是指一个时间序列与其自身在不同时间点上的观测值之间的相关性。对于Durbin - Watson检验，它主要衡量的是回归模型中误差项的自相关程度。\n",
    "- **取值与意义**\n",
    "    - **取值范围**：通常介于 -1到1之间。\n",
    "    - **正值情况**：当`Autocorrelation`为正值时，说明时间序列在相应滞后阶数下，当前观测值与滞后观测值呈现正相关关系，即如果当前值较高，那么滞后的值也倾向于较高；当前值较低，滞后的值也倾向于较低。\n",
    "    - **负值情况**：若`Autocorrelation`为负值，则表示当前观测值与滞后观测值呈负相关关系，即当前值较高时，滞后的值倾向于较低，反之亦然。\n",
    "    - **接近0的情况**：`Autocorrelation`接近0，意味着在该滞后阶数下，时间序列的当前观测值与滞后观测值之间几乎不存在线性相关性，这符合误差项无自相关的理想情况。\n",
    "\n",
    "`lag`和`Autocorrelation`相结合，可以帮助分析人员更全面地了解时间序列中自相关的特征和结构。例如，通过观察不同`lag`下`Autocorrelation`的变化，判断自相关是否存在周期性规律，或者确定自相关在哪些滞后阶数上表现得较为显著等，为进一步的模型诊断和改进提供依据。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "id": "9beca7f7-149c-4308-bbe1-5946ee7be005",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<style>\n",
       ".dl-inline {width: auto; margin:0; padding: 0}\n",
       ".dl-inline>dt, .dl-inline>dd {float: none; width: auto; display: inline-block}\n",
       ".dl-inline>dt::after {content: \":\\0020\"; padding-right: .5ex}\n",
       ".dl-inline>dt:not(:first-of-type) {padding-left: .5ex}\n",
       "</style><dl class=dl-inline><dt>Nevada</dt><dd>28</dd><dt>Rhode Island</dt><dd>39</dd></dl>\n"
      ],
      "text/latex": [
       "\\begin{description*}\n",
       "\\item[Nevada] 28\n",
       "\\item[Rhode Island] 39\n",
       "\\end{description*}\n"
      ],
      "text/markdown": [
       "Nevada\n",
       ":   28Rhode Island\n",
       ":   39\n",
       "\n"
      ],
      "text/plain": [
       "      Nevada Rhode Island \n",
       "          28           39 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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JGUjycCIQRChkWJ1PSIIikfTwRCCIQM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",
      "text/plain": [
       "Plot with title \"Q-Q Plot\""
      ]
     },
     "metadata": {
      "image/png": {
       "height": 420,
       "width": 420
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fit <- lm(Murder~Population+Illiteracy+Income+Frost,data=states)\n",
    "qqPlot(fit,main=\"Q-Q Plot\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "id": "3adfa307-7280-4e27-a6b8-d17d3588f333",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "       Murder Population Illiteracy Income Frost\n",
      "Nevada   11.5        590        0.5   5149   188\n",
      "y_hat 3.878958 rs: 7.621042 rstudent: 3.542929 "
     ]
    }
   ],
   "source": [
    "print(states[c(\"Nevada\"),])\n",
    "cat(\"y_hat\",fitted(fit)[\"Nevada\"],\"\")\n",
    "cat(\"rs:\",residuals(fit)[\"Nevada\"],\"\")\n",
    "cat(\"rstudent:\",rstudent(fit)[\"Nevada\"],\"\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b366c5e6-0c6c-46b9-8fe6-ee0c6ae25d3f",
   "metadata": {},
   "source": [
    "如果模型假设正确，即满足正态分布，学生化残差满足自由度为$n-p-1$的$t$分布,其中p为参数个数（包含截距项）"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "id": "2aaa6004-92e4-4900-a53f-1f7b898d46a9",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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kEksAKRVPIRCZOSwl6k369zfQZ0vvwu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\nDyASgAcQCcAD/wcsmJVwUuRhuAAAAABJRU5ErkJggg==",
      "text/plain": [
       "Plot with title \"Distribution of Errors\""
      ]
     },
     "metadata": {
      "image/png": {
       "height": 420,
       "width": 420
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "residplot <- function(fit,nbreaks=10){\n",
    "    z <- rstudent(fit)\n",
    "    hist(z,breaks=nbreaks,freq=FALSE,xlab=\"Student Residual\",main=\"Distribution of Errors\")\n",
    "    rug(jitter(z),col=\"brown\")\n",
    "    curve(dnorm(x,mean=mean(z),sd=sd(z)),add=TRUE,col=\"blue\",lwd=2)#curve隐式的创建的x\n",
    "    lines(density(z)$x,density(z)$y,col=\"red\",lwd=2,lty=2)\n",
    "    legend(\"topright\",legend=c(\"normal curve\",\"kernel density curve\"),lty=1:2,col=c(\"blue\",\"red\"),cex=.7)\n",
    "}\n",
    "residplot(fit)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d2000ded-1f9a-40d2-9fe1-7ad5d538f6e8",
   "metadata": {},
   "source": [
    "**线性**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "id": "9cafb87f-ece0-4348-bebe-5ca5d0637852",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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cFHLFqrXRekOr+PlK/LWyg5/Z1foKNrTwFJPnCyYdcIZ+1yO871QOJSFCRa8AAg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\niEEACYIY9H+ZgLeYI7EYqQAAAABJRU5ErkJggg==",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 420,
       "width": 420
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "crPlots(fit)#成分与残差图"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "59e09e8e-6422-470e-a354-5280abb093fc",
   "metadata": {},
   "source": [
    "在回归分析中，`crPlots(fit)`生成的成分与残差图（Component and Residual Plots，简称CR图）有以下重要作用：\n",
    "\n",
    "#### 检测变量的线性关系\n",
    "- **观察自变量与残差关系**：通过成分与残差图，可以直观地查看每个自变量与残差之间的关系。如果自变量与残差之间呈现出某种系统性的模式，如曲线关系或明显的趋势，这可能意味着自变量与因变量之间的线性假设不成立，提示可能需要对模型进行非线性变换或添加高阶项等改进。\n",
    "- **评估线性拟合效果**：若数据点在成分与残差图中随机分布在一条水平直线周围，说明自变量与因变量之间的线性关系假设是合理的，当前的线性模型能够较好地拟合数据；反之，如果数据点呈现出非随机的分布模式，如弯曲、聚集等，则表明线性模型可能存在不足，需要进一步调整。\n",
    "\n",
    "#### 识别异常值和强影响点\n",
    "- **发现异常数据点**：成分与残差图可以帮助识别数据中的异常值。那些明显偏离其他数据点，位于图形边缘或孤立的数据点可能是异常值，它们可能对回归模型的估计和推断产生较大影响，需要进一步检查这些数据点的准确性和合理性，考虑是否需要对其进行特殊处理或从数据集中剔除。\n",
    "- **确定强影响点**：除了异常值，成分与残差图还能帮助找出强影响点。强影响点是指那些对回归模型的参数估计有较大影响力的数据点，即使它们不一定是数值上的异常值。在成分与残差图中，强影响点通常表现为与其他数据点有明显不同的分布特征，或者对拟合线的形状和位置有较大的改变作用。\n",
    "\n",
    "#### 诊断模型的有效性\n",
    "- **检查模型假设**：成分与残差图可以用于检查回归模型的一些基本假设是否满足，如误差项的独立性、同方差性等。如果残差在不同自变量取值下呈现出明显的方差变化趋势，即残差的散布宽度在图形的不同位置不一致，这可能暗示存在异方差问题，需要对模型进行相应的调整，如采用加权最小二乘法等方法来修正。\n",
    "- **评估模型拟合优度**：通过观察成分与残差图中数据点与拟合线的接近程度，可以直观地评估模型的拟合优度。数据点越紧密地聚集在拟合线周围，说明模型对数据的拟合效果越好，拟合优度越高；反之，如果数据点与拟合线之间存在较大的偏差和离散度，则表明模型的拟合效果不佳，需要考虑引入更多的自变量、变换变量形式或选择更合适的模型等。\n",
    "\n",
    "#### 指导模型改进和变量选择\n",
    "- **提示变量添加或删除**：成分与残差图可以为变量选择提供有用的信息。如果某个自变量的成分与残差图显示出残差存在明显的可解释模式，可能意味着该自变量对因变量的解释还不够充分，需要考虑添加其他相关变量来进一步完善模型；相反，如果某个自变量的成分与残差图显示出该自变量与残差之间没有明显的关系，或者该自变量对残差的解释能力很弱，可能说明该自变量对模型的贡献不大，可以考虑将其从模型中删除，以简化模型。\n",
    "- **引导变量变换**：根据成分与残差图中呈现的自变量与残差的关系，还可以指导对自变量进行适当的变换。例如，如果发现自变量与残差之间存在非线性关系，可以尝试对自变量进行对数变换、平方变换等，然后重新绘制成分与残差图，观察变换后的效果，以找到更合适的变量形式，提高模型的拟合精度。\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "id": "b58e268d-b18e-4265-92c5-a87cba35bede",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<style>\n",
       ".dl-inline {width: auto; margin:0; padding: 0}\n",
       ".dl-inline>dt, .dl-inline>dd {float: none; width: auto; display: inline-block}\n",
       ".dl-inline>dt::after {content: \":\\0020\"; padding-right: .5ex}\n",
       ".dl-inline>dt:not(:first-of-type) {padding-left: .5ex}\n",
       "</style><dl class=dl-inline><dt>Population</dt><dd>1.24528200205236</dd><dt>Illiteracy</dt><dd>2.16584830171514</dd><dt>Income</dt><dd>1.34582173068518</dd><dt>Frost</dt><dd>2.08254682072994</dd></dl>\n"
      ],
      "text/latex": [
       "\\begin{description*}\n",
       "\\item[Population] 1.24528200205236\n",
       "\\item[Illiteracy] 2.16584830171514\n",
       "\\item[Income] 1.34582173068518\n",
       "\\item[Frost] 2.08254682072994\n",
       "\\end{description*}\n"
      ],
      "text/markdown": [
       "Population\n",
       ":   1.24528200205236Illiteracy\n",
       ":   2.16584830171514Income\n",
       ":   1.34582173068518Frost\n",
       ":   2.08254682072994\n",
       "\n"
      ],
      "text/plain": [
       "Population Illiteracy     Income      Frost \n",
       "  1.245282   2.165848   1.345822   2.082547 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "vif(fit)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f5629af4-55a6-42a7-9574-bbe9f6988b81",
   "metadata": {},
   "source": [
    "一般来说，$VIF$值大于 5 或 10 时，表明存在严重的多重共线性问题，需要对模型中的自变量进行调整，如删除一些高度相关的变量、进行变量变换等。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d3f841f5-658a-4178-967e-9b860c8eb38b",
   "metadata": {},
   "source": [
    "**检验同方差性**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "id": "ba1a4944-dfaf-4f77-8301-f000ed1af3f7",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Non-constant Variance Score Test \n",
       "Variance formula: ~ fitted.values \n",
       "Chisquare = 1.746514, Df = 1, p = 0.18632"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "ncvTest(fit)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "id": "49d59825-e24b-48f0-99e1-dec51c87a5d6",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "Suggested power transformation:  1.209626 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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R6Q3RVThPYjtqgDLZHSXRmISJNo/EV13Ph+FMxsGEnKa9URaQKNvyj/3x4ijSTl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\n/wFuxrglF574jQAAAABJRU5ErkJggg==",
      "text/plain": [
       "Plot with title \"Spread-Level Plot for\n",
       " fit\""
      ]
     },
     "metadata": {
      "image/png": {
       "height": 420,
       "width": 420
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "spreadLevelPlot(fit)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "id": "130d4b6c-253f-4726-b8d0-cbf9574af07e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "Suggested power transformation:  1.027591 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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CtTXeN9xU8/R70twv7EYOJn/EJmiBTa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\n67rfyhQAAAAASUVORK5CYII=",
      "text/plain": [
       "Plot with title \"Spread-Level Plot for\n",
       " fit1\""
      ]
     },
     "metadata": {
      "image/png": {
       "height": 420,
       "width": 420
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fit1 <- lm(I(Murder^1.209626)~Population+Illiteracy+Income+Frost,data=states)\n",
    "spreadLevelPlot(fit1)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f821ac93-a5c5-4565-9098-d9c72013768b",
   "metadata": {},
   "source": [
    "满足同方差性，虚线是一条水平直线，否则不是水平的，并且点在水平虚线的周围随机分布。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "id": "e7accace-110f-4732-8d27-bcb58ac542dd",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "               _                                \n",
       "platform       x86_64-w64-mingw32               \n",
       "arch           x86_64                           \n",
       "os             mingw32                          \n",
       "crt            ucrt                             \n",
       "system         x86_64, mingw32                  \n",
       "status                                          \n",
       "major          4                                \n",
       "minor          4.2                              \n",
       "year           2024                             \n",
       "month          10                               \n",
       "day            31                               \n",
       "svn rev        87279                            \n",
       "language       R                                \n",
       "version.string R version 4.4.2 (2024-10-31 ucrt)\n",
       "nickname       Pile of Leaves                   "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "version"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "id": "3461a96a-a957-4364-a83b-126467895b06",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "Suggested power transformation:  9.078244 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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5WwC1psmLw6RQqBR6QQ0GLD5MUX6Zi1v3FK2AUtNkxerSLVd+yrquzh8cZD1v7G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",
      "text/plain": [
       "Plot with title \"Spread-Level Plot for\n",
       " fit2\""
      ]
     },
     "metadata": {
      "image/png": {
       "height": 420,
       "width": 420
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "currentRow <- !(rownames(states) %in% c(\"Alaska\",\"Rhode Island\"))\n",
    "fit2 <- lm(I(Murder^0.1)~Population+Illiteracy+Income+Frost,data=states)\n",
    "spreadLevelPlot(fit2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "id": "b68c5799-fb63-4199-a4e7-044d74424179",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "Suggested power transformation:  0.4542492 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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ZTI68zH4mchSAhfQCZLpOtLkBC+gEyW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",
      "text/plain": [
       "Plot with title \"Spread-Level Plot for\n",
       " fit3\""
      ]
     },
     "metadata": {
      "image/png": {
       "height": 420,
       "width": 420
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "currentRow <- !(rownames(states) %in% c(\"Alaska\",\"Rhode Island\"))\n",
    "fit3<- lm(I(Murder^2.8)~Population+Illiteracy+Income+Frost,data=states)\n",
    "spreadLevelPlot(fit3)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ae6118cf-d211-4351-a6cd-e1ddb8f6bf51",
   "metadata": {},
   "source": [
    "**线性模型的综合假设**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "id": "69acd94c-7bda-422a-bffd-55c76b9fe527",
   "metadata": {},
   "outputs": [],
   "source": [
    "# install.packages(\"gvlma\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "id": "f14ee53d-4ab1-436a-b175-25c23b7b6c57",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "Call:\n",
      "lm(formula = Murder ~ Population + Illiteracy + Income + Frost, \n",
      "    data = states)\n",
      "\n",
      "Residuals:\n",
      "    Min      1Q  Median      3Q     Max \n",
      "-4.7960 -1.6495 -0.0811  1.4815  7.6210 \n",
      "\n",
      "Coefficients:\n",
      "             Estimate Std. Error t value Pr(>|t|)    \n",
      "(Intercept) 1.235e+00  3.866e+00   0.319   0.7510    \n",
      "Population  2.237e-04  9.052e-05   2.471   0.0173 *  \n",
      "Illiteracy  4.143e+00  8.744e-01   4.738 2.19e-05 ***\n",
      "Income      6.442e-05  6.837e-04   0.094   0.9253    \n",
      "Frost       5.813e-04  1.005e-02   0.058   0.9541    \n",
      "---\n",
      "Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1\n",
      "\n",
      "Residual standard error: 2.535 on 45 degrees of freedom\n",
      "Multiple R-squared:  0.567,\tAdjusted R-squared:  0.5285 \n",
      "F-statistic: 14.73 on 4 and 45 DF,  p-value: 9.133e-08\n",
      "\n",
      "\n",
      "ASSESSMENT OF THE LINEAR MODEL ASSUMPTIONS\n",
      "USING THE GLOBAL TEST ON 4 DEGREES-OF-FREEDOM:\n",
      "Level of Significance =  0.05 \n",
      "\n",
      "Call:\n",
      " gvlma(x = fit) \n",
      "\n",
      "                    Value p-value                Decision\n",
      "Global Stat        2.7728  0.5965 Assumptions acceptable.\n",
      "Skewness           1.5374  0.2150 Assumptions acceptable.\n",
      "Kurtosis           0.6376  0.4246 Assumptions acceptable.\n",
      "Link Function      0.1154  0.7341 Assumptions acceptable.\n",
      "Heteroscedasticity 0.4824  0.4873 Assumptions acceptable.\n"
     ]
    }
   ],
   "source": [
    "library(gvlma)\n",
    "gvmodel <- gvlma(fit)\n",
    "summary(gvmodel)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ce7a8d49-8635-4e34-aadd-41b1b4e95ac5",
   "metadata": {},
   "source": [
    "**线性模型假设的评估**\n",
    "\n",
    "使用4自由度的全局检验：\n",
    "显著性水平 = 0.05\n",
    "\n",
    "调用：\n",
    "gvlma(x = fit) \n",
    "\n",
    "|检验项|值|p值|决策|\n",
    "|----|----|----|----|\n",
    "|全局统计量|2.7728|0.5965|假设可接受|\n",
    "|偏度|1.5374|0.2150|假设可接受|\n",
    "|峰度|0.6376|0.4246|假设可接受|\n",
    "|链接函数|0.1154|0.7341|假设可接受|\n",
    "|异方差性|0.4824|0.4873|假设可接受|\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "id": "27ee75c9-5f8b-43d8-90a8-df70deeef72c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "2.7728"
      ],
      "text/latex": [
       "2.7728"
      ],
      "text/markdown": [
       "2.7728"
      ],
      "text/plain": [
       "[1] 2.7728"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "1.5374+0.6376+0.1154+0.4824"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "id": "00de6ba5-61bb-4833-bd11-863e167fdaf4",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "       rstudent unadjusted p-value Bonferroni p\n",
       "Nevada 3.542929         0.00095088     0.047544"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "outlierTest(fit)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bcbff83c-987f-44ad-87c4-163bc7ce24c9",
   "metadata": {},
   "source": [
    "**高杠杆值**\n",
    "\n",
    "在统计学和回归分析中，“帽子值”（Hat Value）也被称为“杠杆值”（Leverage Value），具有特定的含义和作用。\n",
    "\n",
    "- **定义**：帽子值用于衡量每个观测值对回归模型拟合值的影响程度，它反映了观测值在自变量空间中的位置对回归结果的潜在影响力。\n",
    "- **计算**：对于多元线性回归模型$y = X\\beta+\\epsilon$，其中$y$是响应变量向量，$X$是设计矩阵，$\\beta$是回归系数向量，$\\epsilon$是误差向量。帽子矩阵$H = X(X^{T}X)^{-1}X^{T}$，帽子值$h_{ii}$就是帽子矩阵$H$的对角线上的元素。\n",
    "- 帽子平均值为$\\frac{p}{n}$,$p$为参数个数，当帽子值大于帽子均值的2或3倍，就可以认定为高杠杆点。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "id": "63861e8a-4e86-44a9-a505-c3442a8c97f0",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1] 50\n"
     ]
    },
    {
     "data": {
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bO5fSffWghSeY8ndpuxRozH3f713s/uuWcw569yLoUgHb1fx9+Zx7q338zR15sg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",
      "text/plain": [
       "Plot with title \"index plot of hat values\""
      ]
     },
     "metadata": {
      "image/png": {
       "height": 420,
       "width": 420
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "hatPlot <- function(fit){\n",
    "    p <- length(coefficients(fit))\n",
    "    n <- length(fitted(fit))\n",
    "    print(n)\n",
    "    hatVal <- hatvalues(fit)\n",
    "    plot(hatVal,main=\"index plot of hat values\")\n",
    "    avgHat = p/n\n",
    "    abline(h=c(2,3)*avgHat,col=\"red\",lty=2)\n",
    "    # identify(1:n,hatVal,names(hatVal))\n",
    "    labelIdx = hatVal>2*avgHat\n",
    "    text(c(1:n)[labelIdx],hatVal[labelIdx]-0.01,names(hatVal)[labelIdx],col=\"dark red\")\n",
    "}\n",
    "hatPlot(fit)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9733974c-3ef8-486a-b418-5184e2df7987",
   "metadata": {},
   "source": [
    "**强影响点**\n",
    "\n",
    "检测强影响点的方法：\n",
    "- Cook距离，或称D统计量：当Cook's D值大于4/(n-k-1)则表明它是强影响点，n是样本数量，k是预测变量数目；\n",
    "- 变量添加图(added variable plot)。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "id": "d27fc64d-e6d0-450d-8db6-509b1a0465ec",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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pBS1b0Z21wk8i9FE1t5SdonL0I6Jlap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\nUyAv7H4pJMAiQgIEEBIg4H8MvAHxvwP/ZQAAAABJRU5ErkJggg==",
      "text/plain": [
       "Plot with title \"\""
      ]
     },
     "metadata": {
      "image/png": {
       "height": 420,
       "width": 420
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "cutoff <- 4/(nrow(states)-length(fit$coefficients)-2)\n",
    "plot(fit,which=4,cook.levels=cutoff)\n",
    "abline(h=cutoff,lty=2,col=\"red\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "003d0a07-ee8d-4bb6-aae5-3087ca34785b",
   "metadata": {},
   "source": [
    "**变量添加图**（Added Variable Plot，AvPlot）也称为偏回归图，其绘图原理如下：\n",
    "\n",
    "#### 基于回归分析的残差关系\n",
    "- 对于一个包含多个自变量的线性回归模型$y = \\beta_0+\\beta_1x_1+\\beta_2x_2+\\cdots+\\beta_nx_n+\\epsilon$，假设要绘制自变量$x_i$的变量添加图。\n",
    "- 首先，将因变量$y$对除$x_i$之外的其他所有自变量进行回归，得到回归方程$\\hat{y}_{-i}=\\hat{\\beta}_{0}+\\hat{\\beta}_{1}x_1+\\cdots+\\hat{\\beta}_{i - 1}x_{i - 1}+\\hat{\\beta}_{i + 1}x_{i + 1}+\\cdots+\\hat{\\beta}_{n}x_{n}$，并计算出残差$e_y = y-\\hat{y}_{-i}$。\n",
    "- 然后，将自变量$x_i$也对除它自身之外的其他自变量进行回归，得到$\\hat{x}_{i}=\\hat{\\alpha}_{0}+\\hat{\\alpha}_{1}x_1+\\cdots+\\hat{\\alpha}_{i - 1}x_{i - 1}+\\hat{\\alpha}_{i + 1}x_{i + 1}+\\cdots+\\hat{\\alpha}_{n}x_{n}$，并计算出残差$e_{x_i}=x_i-\\hat{x}_{i}$。\n",
    "\n",
    "#### 绘制原理\n",
    "- **横纵坐标**：以$e_{x_i}$为横坐标，$e_y$为纵坐标绘制散点图。这样做的目的是在剔除了其他自变量对$y$和$x_i$的线性影响后，单独观察$x_i$与$y$之间的关系。\n",
    "- **回归直线**：在散点图上，通常还会添加一条拟合直线，该直线的斜率就是在控制了其他自变量的情况下，$x_i$对$y$的偏回归系数$\\hat{\\beta}_i$的估计值。这条直线展示了在排除其他变量干扰后，$x_i$与$y$之间的线性关系趋势。\n",
    "\n",
    "#### 原理意义\n",
    "- **揭示变量真实关系**：通过这种方式绘制的变量添加图，可以更清晰地展示某个自变量与因变量之间的真实关系，避免了由于其他自变量的存在而可能产生的混淆或假象。例如，在研究农作物产量与施肥量、降雨量、光照时间等多个因素的关系时，变量添加图可以帮助准确看出施肥量与产量之间的关系，而不受降雨量和光照时间的干扰。\n",
    "- **检测共线性**：如果自变量之间存在严重的共线性，那么在变量添加图中，散点的分布会呈现出一些特殊的模式，如散点云呈带状分布且拟合直线的斜率不稳定等。这是因为共线性会导致自变量之间的信息重叠，使得在剔除其他自变量影响后，自变量与因变量的关系变得不稳定。\n",
    "- **评估变量重要性**：从变量添加图中拟合直线的斜率大小和显著性，可以直观地评估每个自变量对因变量的相对重要性。斜率绝对值越大，说明该自变量对因变量的影响越大，在模型中的重要性可能越高。\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "id": "e641ea51-661a-44c6-b683-9bbd3e4a562f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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NRWbpASMbJB2rwDi8O126ggBbMvEECqZWIxkAL2nSAlc+S+ICv0d1cMQvte28cA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\nBDEIIEEQgwASBDEIIEEQg/4fOaotj6JmISIAAAAASUVORK5CYII=",
      "text/plain": [
       "Plot with title \"\""
      ]
     },
     "metadata": {
      "image/png": {
       "height": 420,
       "width": 420
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "avPlots(fit)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "id": "94b6a200-9dca-4458-86ff-5630e5c5d7ed",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "                StudRes        Hat       CookD\n",
      "Alaska        1.7536917 0.43247319 0.448050997\n",
      "California   -0.2761492 0.34087628 0.008052956\n",
      "Nevada        3.5429286 0.09508977 0.209915743\n",
      "Rhode Island -2.0001631 0.04562377 0.035858963\n"
     ]
    },
    {
     "data": {
      "image/png": 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8pfwchLsMtkMt/9znfjX48njibC/eGB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\nADTx/7K4oNV3ZJBtAAAAAElFTkSuQmCC",
      "text/plain": [
       "Plot with title \"Influence Plot\""
      ]
     },
     "metadata": {
      "image/png": {
       "height": 420,
       "width": 420
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "print(influencePlot(fit,main=\"Influence Plot\",id=list(method=\"noteworthy\") ))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0a69cb4f-c433-4105-bd7b-c31affb02f63",
   "metadata": {},
   "source": [
    "纵坐标超过+2或者小于-2可被认定为离群点，水平轴超过0.2或0.3的州有高杠杆值。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "15210d21-4cfe-40be-8021-4295d0ab2f0e",
   "metadata": {},
   "source": [
    "**Box - Cox变换**是一种用于对数据进行变换的统计方法，由统计学家乔治·博克斯（George Box）和德里克·考克斯（Derek Cox）提出，以下是其详细介绍：\n",
    "\n",
    "#### 定义与公式\n",
    "- Box - Cox变换的一般公式为：$y^{$lambda)}=\\begin{cases}\\frac{y^{\\lambda}-1}{\\lambda},&\\lambda\\neq0\\\\\\ln y,&\\lambda = 0\\end{cases}$，其中$y$是原始数据，$y^{$lambda)}$是变换后的数据，$\\lambda$是变换参数。\n",
    "\n",
    "#### 目的与作用\n",
    "- **使数据满足正态性假设**：在许多统计分析中，通常要求数据服从正态分布。但实际数据往往不满足这一条件，通过Box - Cox变换，可以将非正态数据转换为更接近正态分布的数据，从而使基于正态分布假设的统计方法（如线性回归、方差分析等）更加有效和可靠。\n",
    "- **稳定数据方差**：一些数据可能存在方差不齐的问题，即不同组或不同水平的数据方差差异较大。Box - Cox变换可以在一定程度上使数据的方差保持稳定，这对于一些需要方差齐性假设的统计方法非常重要。\n",
    "\n",
    "#### 变换参数$\\lambda$的确定\n",
    "- **最大似然估计法**：通过构建似然函数，找到使似然函数最大的$\\lambda$值。在实际应用中，通常使用统计软件来计算最大似然估计的$\\lambda$值。\n",
    "- **交叉验证法**：将数据集划分为训练集和验证集，在训练集上使用不同的$\\lambda$值进行Box - Cox变换并建立模型，然后在验证集上评估模型的性能，选择使模型性能最优的$\\lambda$值。\n",
    "\n",
    "#### 应用场景\n",
    "- **经济数据分析**：在分析经济指标（如收入、消费等）时，Box - Cox变换可将数据转换为更适合分析的形式，帮助经济学家更好地建立经济模型，进行预测和政策分析。\n",
    "- **生物医学研究**：在处理生物实验数据（如药物浓度、生物指标等）时，该变换可使数据满足统计分析的假设条件，提高研究结果的准确性和可靠性。\n",
    "- **工程与质量控制**：在工程领域，用于处理一些非正态分布的质量数据，如产品的尺寸误差、强度等，使数据更符合统计过程控制和质量分析的要求。\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "id": "a6abba4b-7bab-43d4-be35-e7867701ed54",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "bcPower Transformation to Normality \n",
       "              Est Power Rounded Pwr Wald Lwr Bnd Wald Upr Bnd\n",
       "states$Murder    0.6055           1       0.0884       1.1227\n",
       "\n",
       "Likelihood ratio test that transformation parameter is equal to 0\n",
       " (log transformation)\n",
       "                           LRT df     pval\n",
       "LR test, lambda = (0) 5.665991  1 0.017297\n",
       "\n",
       "Likelihood ratio test that no transformation is needed\n",
       "                           LRT df    pval\n",
       "LR test, lambda = (1) 2.122763  1 0.14512"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "summary(powerTransform(states$Murder))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b5d44db7-f8c4-47da-8cdf-a1ec86f81675",
   "metadata": {},
   "source": [
    "以下是对上述 `powerTransform(states$Murder)` 函数返回结果的详细解读：\n",
    "\n",
    "#### “bcPower Transformation to Normality”部分\n",
    "- **“Est Power”（估计幂次）**：这里显示的值为 `0.6055`，它表示通过相应的算法（比如最大似然估计等方法）估计出来的进行幂变换的参数 `λ` 的值。这个值提示了对于 `states$Murder` 这个变量（可能是某个州的谋杀数据相关变量），按照 `Box - Cox` 变换的公式，进行幂变换时建议使用的 `λ` 参数大小。\n",
    "- **“Rounded Pwr”（取整后的幂次）**：其值为 `1`，意味着在实际应用中，经过一定的取整规则后，建议采用的幂次为 `1` 。这可能是考虑到实际操作便利性以及数据解读等因素，对估计出来的 `0.6055` 进行了取整处理，当取整为 `1` 时，对应的变换就类似于不对数据做特殊的幂变换（不过严格来说，按照 `Box - Cox` 公式，`λ = 1` 时也有对应的数学变换形式）。\n",
    "- **“Wald Lwr Bnd”（沃尔德下限边界）和“Wald Upr Bnd”（沃尔德上限边界）**：分别为 `0.0884` 和 `1.1227`，这两个值构成了通过沃尔德检验（一种统计检验方法）得到的关于估计幂次 `λ` 的置信区间边界。它表示在一定的置信水平下（通常是常规的统计置信水平），真实的 `λ` 值有较大概率落在这个区间范围内。\n",
    "\n",
    "#### “Likelihood ratio test that transformation parameter is equal to 0 (log transformation)”部分\n",
    "- **“LRT”（似然比检验统计量）**：值为 `5.665991`，似然比检验（Likelihood Ratio Test，简称 LRT）在这里是用于检验变换参数 `λ` 是否等于 `0` 的假设。当 `λ = 0` 时，按照 `Box - Cox` 变换公式，对应的就是对数据进行自然对数变换（`ln y`）。\n",
    "- **“df”（自由度）**：这里自由度为 `1`，它取决于检验所涉及的参数个数等因素，在这个单参数（即 `λ`）的检验场景下，自由度通常为 `1`。\n",
    "- **“pval”（p 值）**：值为 `0.017297`，在统计学中，`p` 值用于判断是否拒绝原假设。一般来说，如果 `p` 值小于预先设定的显著性水平（比如常用的 `0.05`），则拒绝原假设。在此处，原假设是 `λ = 0`（即进行对数变换是合适的），由于 `0.017297 < 0.05`，所以从这个检验结果来看，有足够的证据拒绝原假设，意味着单纯的对数变换可能不是最适合使该变量数据达到正态性的变换方式。\n",
    "\n",
    "#### “Likelihood ratio test that no transformation is needed”部分\n",
    "- **“LRT”**：值为 `2.122763`，这个似然比检验是用于检验是否不需要进行任何变换（也就是 `λ = 1` 的情况，可理解为原始数据形式就合适）的假设。\n",
    "- **“df”**：同样为 `1`，基于和前面类似的自由度确定逻辑。\n",
    "- **“pval”**：值为 `0.14512`，原假设是 `λ = 1`（即不需要变换），由于 `0.14512 > 0.05`，按照常规的显著性水平判断标准，没有足够的证据拒绝原假设，说明从这个检验角度来看，也不能完全排除原始数据形式（不进行变换）是可以接受的一种情况，但这个证据相对来说不是很强，只是没有强到能拒绝原始数据形式合适这一假设的程度。\n",
    "\n",
    "总体而言，这些结果综合起来是在通过不同的检验和估计方式来探讨对 `states$Murder` 变量进行何种 `Box - Cox` 变换（或者是否需要变换）更有利于使其数据符合正态分布假设，不同的检验结果从不同侧面给出了参考信息，以帮助使用者做出决策。 "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3fd7c7f3-ef3b-47ba-a78b-988aa30c6ac1",
   "metadata": {},
   "source": [
    "**[计量经济学的三大检验](https://blog.csdn.net/u011517132/article/details/105565678/)**\n",
    "- 似然比检验（LR）\n",
    "- 沃尔德检验（Wald）\n",
    "- 拉格朗日乘子检验（LM）"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "id": "b9157a91-f2a3-40b0-a9a8-4de22a001d59",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "           MLE of lambda Score Statistic (t) Pr(>|t|)\n",
       "Population       0.86939             -0.3228   0.7483\n",
       "Illiteracy       1.35812              0.6194   0.5388\n",
       "\n",
       "iterations =  19 \n",
       "\n",
       "Score test for null hypothesis that all lambdas = 1:\n",
       "F = 0.23214, df = 2 and 45, Pr(>F) = 0.7938\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "boxTidwell(Murder~Population+Illiteracy,data=states)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bdc086d0-347b-4024-9990-768fe600f43e",
   "metadata": {},
   "source": [
    "以下是对 `boxTidwell(Murder~Population+Illiteracy, data=states)` 结果的详细解读：\n",
    "\n",
    "#### “MLE of lambda”（lambda的最大似然估计）部分\n",
    "- **“Population”行**：对于自变量“Population”（可能表示人口相关变量），其对应的 `lambda` 的最大似然估计值为 `0.86939`。这意味着在 `Box - Tidwell` 变换的情境下，根据最大似然估计的方法，建议对该自变量进行幂次为 `0.86939` 的变换，以更好地满足回归模型等相关统计分析的一些假设条件（比如线性关系、正态性等假设）。\n",
    "- **“Illiteracy”行**：自变量“Illiteracy”（可能是文盲率相关变量）对应的 `lambda` 的最大似然估计值为 `1.35812`，提示对该自变量按此幂次进行变换可能有助于优化统计分析效果。\n",
    "\n",
    "#### “Score Statistic (t)”（得分统计量（t值））与“Pr(>|t|)”（t检验的p值）部分\n",
    "- **“Population”行**：\n",
    "    - “Score Statistic (t)”的值为 `-0.3228`，得分统计量（Score Statistic）常与似然函数相关联，用于检验特定的统计假设，在这里是和 `lambda` 的估计值相关的一种检验统计量。\n",
    "    - “Pr(>|t|)”的值为 `0.7483`，这是对应于上述 `t` 值的 `p` 值。在假设检验中，通常以一个预先设定的显著性水平（如 `0.05`）来判断是否拒绝原假设。此处原假设一般是指当前估计的 `lambda` 值所对应的情况与某个基准假设（比如 `lambda = 1`，意味着不需要进行变换的情况）没有差异。由于 `0.7483 > 0.05`，所以没有足够的证据拒绝原假设，也就是说，从这个检验角度来看，对于自变量“Population”，目前没有很强的证据表明需要对其进行不同于原始形式（即类似 `lambda = 1` 情况）的变换来满足模型假设等要求。\n",
    "- **“Illiteracy”行**：\n",
    "    - “Score Statistic (t)”的值为 `0.6194`，同样是基于得分统计量的计算结果。\n",
    "    - “Pr(>|t|)”的值为 `0.5388`，鉴于 `0.5388 > 0.05`，和“Population”自变量类似，对于自变量“Illiteracy”，也没有足够有力的证据说明一定要按照估计的 `lambda` 值（`1.35812`）进行变换，不能拒绝原假设，原始形式（接近 `lambda = 1` 情况）在当前检验下也有一定合理性。\n",
    "\n",
    "#### “iterations = 19”\n",
    "这表示在进行计算以获得上述最大似然估计等结果的过程中，迭代运算的次数为 `19` 次。一般来说，迭代是逐步逼近最优解的过程，在这里就是不断调整参数估计值以达到最大似然估计的较优结果。\n",
    "\n",
    "#### “Score test for null hypothesis that all lambdas = 1: F = 0.23214, df = 2 and 45, Pr(>F) = 0.7938”部分\n",
    "- **“Score test”（得分检验）**：这是一种检验方式，此处用于检验所有自变量对应的 `lambda` 值都等于 `1` 这个原假设，也就是检验是否所有自变量都不需要进行 `Box - Tidwell` 变换，原始数据形式就满足模型要求的假设情况。\n",
    "- **“F”统计量与自由度**：“F = 0.23214”是得分检验计算得到的 `F` 统计量的值，它是衡量不同组间变异程度等因素的一个综合统计量。“df = 2 and 45”表示自由度分别为 `2` 和 `45`，其中 `2` 通常与自变量的个数有关（这里有“Population”和“Illiteracy”两个自变量），`45` 则与样本量等因素相关，在计算 `F` 统计量的过程中确定下来。\n",
    "- **“Pr(>F)”（p值）**：值为 `0.7938`，由于这个 `p` 值大于常规的显著性水平（如 `0.05`），所以从这个整体检验来看，没有足够的证据拒绝原假设，意味着综合考虑这两个自变量，目前的数据没有强烈显示出需要进行 `Box - Tidwell` 变换来改变自变量形式以满足模型假设等情况，原始数据形式在一定程度上是可以接受的。\n",
    "\n",
    "总体而言，这些结果是在通过 `Box - Tidwell` 变换相关的统计检验和估计方法，来分析自变量“Population”和“Illiteracy”是否需要进行特定幂次变换，以更好地适配包含因变量“Murder”的回归模型等统计分析场景，从当前的检验结果来看，没有非常强有力的证据表明一定要进行变换，原始数据形式具有一定的合理性，但也不能完全排除进行变换可能带来更好效果的可能性，具体还需要结合更多的分析和实际应用场景来综合判断。 "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "id": "668e8c88-61e8-4f78-945f-496a61d0ad26",
   "metadata": {},
   "outputs": [
    {
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       "</style><dl class=dl-inline><dt>Alabama</dt><dd>10.9882078964162</dd><dt>Alaska</dt><dd>8.02566023380976</dd><dt>Arizona</dt><dd>9.48700264493595</dd><dt>Arkansas</dt><dd>9.8333194382487</dd><dt>California</dt><dd>10.8742479188743</dd><dt>Colorado</dt><dd>5.11405506874453</dd><dt>Connecticut</dt><dd>6.91042204156619</dd><dt>Delaware</dt><dd>5.46231723427746</dd><dt>Florida</dt><dd>8.78821120962928</dd><dt>Georgia</dt><dd>10.9216195647207</dd><dt>Hawaii</dt><dd>9.61984293644078</dd><dt>Idaho</dt><dd>4.24072327412677</dd><dt>Illinois</dt><dd>7.870452072071</dd><dt>Indiana</dt><dd>5.68106079267724</dd><dt>Iowa</dt><dd>4.32545719745947</dd><dt>Kansas</dt><dd>4.59731294267386</dd><dt>Kentucky</dt><dd>8.91505890888769</dd><dt>Louisiana</dt><dd>13.921175507117</dd><dt>Maine</dt><dd>4.70277259136457</dd><dt>Maryland</dt><dd>6.28520451786973</dd><dt>Massachusetts</dt><dd>7.4583461139229</dd><dt>Michigan</dt><dd>7.37976747192111</dd><dt>Minnesota</dt><dd>4.99149080695137</dd><dt>Mississippi</dt><dd>11.9296482414205</dd><dt>Missouri</dt><dd>5.95193678997685</dd><dt>Montana</dt><dd>4.2572837177289</dd><dt>Nebraska</dt><dd>4.43684814119753</dd><dt>Nevada</dt><dd>3.87895839633072</dd><dt>New Hampshire</dt><dd>4.69312272525265</dd><dt>New Jersey</dt><dd>7.83613735425164</dd><dt>New Mexico</dt><dd>10.9064385331859</dd><dt>New York</dt><dd>11.4412317617713</dd><dt>North Carolina</dt><dd>10.2048370541195</dd><dt>North Dakota</dt><dd>5.12716517263043</dd><dt>Ohio</dt><dd>7.31591059432727</dd><dt>Oklahoma</dt><dd>6.70323289577556</dd><dt>Oregon</dt><dd>4.5569364341483</dd><dt>Pennsylvania</dt><dd>8.390059733848</dd><dt>Rhode Island</dt><dd>7.19596632897292</dd><dt>South Carolina</dt><dd>11.6649260248972</dd><dt>South Dakota</dt><dd>3.82674690834608</dd><dt>Tennessee</dt><dd>9.49764105277539</dd><dt>Texas</dt><dd>13.3760756008256</dd><dt>Utah</dt><dd>4.32810182834192</dd><dt>Vermont</dt><dd>4.17520677314853</dd><dt>Virginia</dt><dd>8.50093308480522</dd><dt>Washington</dt><dd>4.84828948736645</dd><dt>West Virginia</dt><dd>7.72808130448493</dd><dt>Wisconsin</dt><dd>5.53545933052441</dd><dt>Wyoming</dt><dd>4.19909634483974</dd></dl>\n"
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       "\\begin{description*}\n",
       "\\item[Alabama] 10.9882078964162\n",
       "\\item[Alaska] 8.02566023380976\n",
       "\\item[Arizona] 9.48700264493595\n",
       "\\item[Arkansas] 9.8333194382487\n",
       "\\item[California] 10.8742479188743\n",
       "\\item[Colorado] 5.11405506874453\n",
       "\\item[Connecticut] 6.91042204156619\n",
       "\\item[Delaware] 5.46231723427746\n",
       "\\item[Florida] 8.78821120962928\n",
       "\\item[Georgia] 10.9216195647207\n",
       "\\item[Hawaii] 9.61984293644078\n",
       "\\item[Idaho] 4.24072327412677\n",
       "\\item[Illinois] 7.870452072071\n",
       "\\item[Indiana] 5.68106079267724\n",
       "\\item[Iowa] 4.32545719745947\n",
       "\\item[Kansas] 4.59731294267386\n",
       "\\item[Kentucky] 8.91505890888769\n",
       "\\item[Louisiana] 13.921175507117\n",
       "\\item[Maine] 4.70277259136457\n",
       "\\item[Maryland] 6.28520451786973\n",
       "\\item[Massachusetts] 7.4583461139229\n",
       "\\item[Michigan] 7.37976747192111\n",
       "\\item[Minnesota] 4.99149080695137\n",
       "\\item[Mississippi] 11.9296482414205\n",
       "\\item[Missouri] 5.95193678997685\n",
       "\\item[Montana] 4.2572837177289\n",
       "\\item[Nebraska] 4.43684814119753\n",
       "\\item[Nevada] 3.87895839633072\n",
       "\\item[New Hampshire] 4.69312272525265\n",
       "\\item[New Jersey] 7.83613735425164\n",
       "\\item[New Mexico] 10.9064385331859\n",
       "\\item[New York] 11.4412317617713\n",
       "\\item[North Carolina] 10.2048370541195\n",
       "\\item[North Dakota] 5.12716517263043\n",
       "\\item[Ohio] 7.31591059432727\n",
       "\\item[Oklahoma] 6.70323289577556\n",
       "\\item[Oregon] 4.5569364341483\n",
       "\\item[Pennsylvania] 8.390059733848\n",
       "\\item[Rhode Island] 7.19596632897292\n",
       "\\item[South Carolina] 11.6649260248972\n",
       "\\item[South Dakota] 3.82674690834608\n",
       "\\item[Tennessee] 9.49764105277539\n",
       "\\item[Texas] 13.3760756008256\n",
       "\\item[Utah] 4.32810182834192\n",
       "\\item[Vermont] 4.17520677314853\n",
       "\\item[Virginia] 8.50093308480522\n",
       "\\item[Washington] 4.84828948736645\n",
       "\\item[West Virginia] 7.72808130448493\n",
       "\\item[Wisconsin] 5.53545933052441\n",
       "\\item[Wyoming] 4.19909634483974\n",
       "\\end{description*}\n"
      ],
      "text/markdown": [
       "Alabama\n",
       ":   10.9882078964162Alaska\n",
       ":   8.02566023380976Arizona\n",
       ":   9.48700264493595Arkansas\n",
       ":   9.8333194382487California\n",
       ":   10.8742479188743Colorado\n",
       ":   5.11405506874453Connecticut\n",
       ":   6.91042204156619Delaware\n",
       ":   5.46231723427746Florida\n",
       ":   8.78821120962928Georgia\n",
       ":   10.9216195647207Hawaii\n",
       ":   9.61984293644078Idaho\n",
       ":   4.24072327412677Illinois\n",
       ":   7.870452072071Indiana\n",
       ":   5.68106079267724Iowa\n",
       ":   4.32545719745947Kansas\n",
       ":   4.59731294267386Kentucky\n",
       ":   8.91505890888769Louisiana\n",
       ":   13.921175507117Maine\n",
       ":   4.70277259136457Maryland\n",
       ":   6.28520451786973Massachusetts\n",
       ":   7.4583461139229Michigan\n",
       ":   7.37976747192111Minnesota\n",
       ":   4.99149080695137Mississippi\n",
       ":   11.9296482414205Missouri\n",
       ":   5.95193678997685Montana\n",
       ":   4.2572837177289Nebraska\n",
       ":   4.43684814119753Nevada\n",
       ":   3.87895839633072New Hampshire\n",
       ":   4.69312272525265New Jersey\n",
       ":   7.83613735425164New Mexico\n",
       ":   10.9064385331859New York\n",
       ":   11.4412317617713North Carolina\n",
       ":   10.2048370541195North Dakota\n",
       ":   5.12716517263043Ohio\n",
       ":   7.31591059432727Oklahoma\n",
       ":   6.70323289577556Oregon\n",
       ":   4.5569364341483Pennsylvania\n",
       ":   8.390059733848Rhode Island\n",
       ":   7.19596632897292South Carolina\n",
       ":   11.6649260248972South Dakota\n",
       ":   3.82674690834608Tennessee\n",
       ":   9.49764105277539Texas\n",
       ":   13.3760756008256Utah\n",
       ":   4.32810182834192Vermont\n",
       ":   4.17520677314853Virginia\n",
       ":   8.50093308480522Washington\n",
       ":   4.84828948736645West Virginia\n",
       ":   7.72808130448493Wisconsin\n",
       ":   5.53545933052441Wyoming\n",
       ":   4.19909634483974\n",
       "\n"
      ],
      "text/plain": [
       "       Alabama         Alaska        Arizona       Arkansas     California \n",
       "     10.988208       8.025660       9.487003       9.833319      10.874248 \n",
       "      Colorado    Connecticut       Delaware        Florida        Georgia \n",
       "      5.114055       6.910422       5.462317       8.788211      10.921620 \n",
       "        Hawaii          Idaho       Illinois        Indiana           Iowa \n",
       "      9.619843       4.240723       7.870452       5.681061       4.325457 \n",
       "        Kansas       Kentucky      Louisiana          Maine       Maryland \n",
       "      4.597313       8.915059      13.921176       4.702773       6.285205 \n",
       " Massachusetts       Michigan      Minnesota    Mississippi       Missouri \n",
       "      7.458346       7.379767       4.991491      11.929648       5.951937 \n",
       "       Montana       Nebraska         Nevada  New Hampshire     New Jersey \n",
       "      4.257284       4.436848       3.878958       4.693123       7.836137 \n",
       "    New Mexico       New York North Carolina   North Dakota           Ohio \n",
       "     10.906439      11.441232      10.204837       5.127165       7.315911 \n",
       "      Oklahoma         Oregon   Pennsylvania   Rhode Island South Carolina \n",
       "      6.703233       4.556936       8.390060       7.195966      11.664926 \n",
       "  South Dakota      Tennessee          Texas           Utah        Vermont \n",
       "      3.826747       9.497641      13.376076       4.328102       4.175207 \n",
       "      Virginia     Washington  West Virginia      Wisconsin        Wyoming \n",
       "      8.500933       4.848289       7.728081       5.535459       4.199096 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "predict(fit)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "id": "7e4cbcd7-5601-408e-a04c-bbf8dac6f00c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "Call:\n",
       "lm(formula = Murder ~ Population + Illiteracy + Income + Frost, \n",
       "    data = states)\n",
       "\n",
       "Residuals:\n",
       "    Min      1Q  Median      3Q     Max \n",
       "-4.7960 -1.6495 -0.0811  1.4815  7.6210 \n",
       "\n",
       "Coefficients:\n",
       "             Estimate Std. Error t value Pr(>|t|)    \n",
       "(Intercept) 1.235e+00  3.866e+00   0.319   0.7510    \n",
       "Population  2.237e-04  9.052e-05   2.471   0.0173 *  \n",
       "Illiteracy  4.143e+00  8.744e-01   4.738 2.19e-05 ***\n",
       "Income      6.442e-05  6.837e-04   0.094   0.9253    \n",
       "Frost       5.813e-04  1.005e-02   0.058   0.9541    \n",
       "---\n",
       "Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1\n",
       "\n",
       "Residual standard error: 2.535 on 45 degrees of freedom\n",
       "Multiple R-squared:  0.567,\tAdjusted R-squared:  0.5285 \n",
       "F-statistic: 14.73 on 4 and 45 DF,  p-value: 9.133e-08\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "summary(fit)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "id": "c3ad090f-bc58-4833-911a-ae81d62e1fe0",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "              Murder Population Illiteracy Income Frost\n",
      "Vermont          5.5        472        0.6   3907   168\n",
      "Virginia         9.5       4981        1.4   4701    85\n",
      "Washington       4.3       3559        0.6   4864    32\n",
      "West Virginia    6.7       1799        1.4   3617   100\n",
      "Wisconsin        3.0       4589        0.7   4468   149\n",
      "Wyoming          6.9        376        0.6   4566   173\n"
     ]
    }
   ],
   "source": [
    "testSet <- tail(states)\n",
    "print(testSet)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "id": "92ec325d-188c-4f7e-9032-23b69d64d4b9",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<style>\n",
       ".dl-inline {width: auto; margin:0; padding: 0}\n",
       ".dl-inline>dt, .dl-inline>dd {float: none; width: auto; display: inline-block}\n",
       ".dl-inline>dt::after {content: \":\\0020\"; padding-right: .5ex}\n",
       ".dl-inline>dt:not(:first-of-type) {padding-left: .5ex}\n",
       "</style><dl class=dl-inline><dt>Vermont</dt><dd>4.17520677314853</dd><dt>Virginia</dt><dd>8.50093308480522</dd><dt>Washington</dt><dd>4.84828948736645</dd><dt>West Virginia</dt><dd>7.72808130448493</dd><dt>Wisconsin</dt><dd>5.53545933052441</dd><dt>Wyoming</dt><dd>4.19909634483974</dd></dl>\n"
      ],
      "text/latex": [
       "\\begin{description*}\n",
       "\\item[Vermont] 4.17520677314853\n",
       "\\item[Virginia] 8.50093308480522\n",
       "\\item[Washington] 4.84828948736645\n",
       "\\item[West Virginia] 7.72808130448493\n",
       "\\item[Wisconsin] 5.53545933052441\n",
       "\\item[Wyoming] 4.19909634483974\n",
       "\\end{description*}\n"
      ],
      "text/markdown": [
       "Vermont\n",
       ":   4.17520677314853Virginia\n",
       ":   8.50093308480522Washington\n",
       ":   4.84828948736645West Virginia\n",
       ":   7.72808130448493Wisconsin\n",
       ":   5.53545933052441Wyoming\n",
       ":   4.19909634483974\n",
       "\n"
      ],
      "text/plain": [
       "      Vermont      Virginia    Washington West Virginia     Wisconsin \n",
       "     4.175207      8.500933      4.848289      7.728081      5.535459 \n",
       "      Wyoming \n",
       "     4.199096 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "predict(fit,newdata=testSet)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "af1f145d-dfae-422d-afc3-3f650b14813b",
   "metadata": {},
   "source": [
    "突然想起来predict不知道怎么用了，现在知道了"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "id": "725368c6-c73e-412d-8337-77dda1702efd",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Analysis of Variance Table\n",
      "\n",
      "Model 1: Murder ~ Population + Illiteracy + Income + Frost\n",
      "Model 2: Murder ~ Population + Illiteracy\n",
      "  Res.Df    RSS Df Sum of Sq      F Pr(>F)\n",
      "1     45 289.17                           \n",
      "2     47 289.25 -2 -0.078505 0.0061 0.9939\n"
     ]
    }
   ],
   "source": [
    "fit1 <- lm(Murder~Population+Illiteracy+Income+Frost,data=states)\n",
    "fit2 <- lm(Murder~Population+Illiteracy,data=states)\n",
    "print(anova(fit1,fit2))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c55ec2fc-f470-4810-9022-03700f73f3c8",
   "metadata": {},
   "source": [
    "**方差分析(ANOVA)** 可以用于比较两个模型的拟合优度，其基本原理是通过比较模型的方差来判断哪个模型对数据的拟合效果更好。以下以线性回归模型为例，介绍如何通过ANOVA比较两个模型的拟合优度：\n",
    "\n",
    "#### 原理\n",
    "- 对于两个嵌套模型，即一个模型（简化模型）是另一个模型（全模型）的特殊情况，可使用ANOVA进行比较。全模型包含更多的自变量或更复杂的项，简化模型则相对简单。\n",
    "- ANOVA通过比较两个模型的残差平方和（RSS）来判断它们的拟合优度。残差平方和衡量了模型预测值与实际观测值之间的差异，RSS越小，模型拟合效果越好。\n",
    "\n",
    "#### 计算步骤\n",
    "1. **建立模型**\n",
    "    - **全模型**：假设全模型为$y = \\beta_{0}+\\beta_{1}x_{1}+\\beta_{2}x_{2}+\\cdots+\\beta_{p}x_{p}+\\epsilon$，其中$y$是因变量，$x_{1},x_{2},\\cdots,x_{p}$是自变量，$\\beta_{0},\\beta_{1},\\cdots,\\beta_{p}$是回归系数，$\\epsilon$是误差项。\n",
    "    - **简化模型**：简化模型是全模型的一种特殊情况，如$y=\\beta_{0}+\\beta_{1}x_{1}+\\epsilon$，即去掉了全模型中的一些自变量。\n",
    "2. **估计模型参数**\n",
    "    - 使用最小二乘法等方法分别对全模型和简化模型进行参数估计，得到各自模型的回归系数估计值。\n",
    "3. **计算残差平方和**\n",
    "    - 对于全模型，计算其残差平方和$RSS_{F}=\\sum_{i = 1}^{n}(y_{i}-\\hat{y}_{i,F})^{2}$，其中$y_{i}$是第$i$个观测值，$\\hat{y}_{i,F}$是全模型对第$i$个观测值的预测值，$n$是观测值的数量。\n",
    "    - 同理，对于简化模型，计算其残差平方和$RSS_{R}=\\sum_{i = 1}^{n}(y_{i}-\\hat{y}_{i,R})^{2}$，其中$\\hat{y}_{i,R}$是简化模型对第$i$个观测值的预测值。\n",
    "4. **计算F统计量**\n",
    "$F=\\frac{(RSS_{R}-RSS_{F})/(df_{R}-df_{F})}{RSS_{F}/df_{F}}$，其中$df_{R}$和$df_{F}$分别是简化模型和全模型的残差自由度，$df_{R}=n - k_{R}$，$df_{F}=n - k_{F}$，$k_{R}$和$k_{F}$分别是简化模型和全模型中待估计参数的个数。\n",
    "5. **确定显著性水平并进行决策**\n",
    "    - 给定显著性水平$\\alpha$，如$\\alpha = 0.05$。\n",
    "    - 根据$F$统计量和相应的自由度，查找$F$分布表得到临界值$F_{\\alpha}(df_{R}-df_{F},df_{F})$。\n",
    "    - 若$F>F_{\\alpha}(df_{R}-df_{F},df_{F})$，则拒绝原假设，认为全模型的拟合优度显著优于简化模型；反之，则不能拒绝原假设，即没有足够证据表明全模型比简化模型拟合得更好。\n",
    "\n",
    "通过ANOVA比较两个模型的拟合优度时，模型需满足一定的假设条件，如误差项服从正态分布、方差齐性等，否则结果可能不准确。\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "id": "2073b652-8ee8-4054-89b7-5ca1d0f13d8a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "3.20431729211419"
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       "3.20431729211419"
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      ],
      "text/plain": [
       "[1] 3.204317"
      ]
     },
     "metadata": {},
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   ],
   "source": [
    "qf(0.95,2,45)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "id": "0d6bdeb4-2480-4c1d-ade6-8356def95d98",
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      "text/plain": [
       "[1] 0.006224712"
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    }
   ],
   "source": [
    "(289.25-289.17)/2/289.17*45"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "id": "092589c9-4470-44a7-b7ec-27859800c433",
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   "outputs": [
    {
     "data": {
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      "text/plain": [
       "[1] 0.9937955"
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     "metadata": {},
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    }
   ],
   "source": [
    "1-pf(0.00622471210706381,2,45)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "id": "562ebf9b-58fb-4cfa-96a5-1a11ef61d9c2",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "     df      AIC\n",
      "fit1  6 241.6429\n",
      "fit2  4 237.6565\n"
     ]
    }
   ],
   "source": [
    "print(AIC(fit1,fit2))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f1e33c5d-f175-40f2-a223-27860f339625",
   "metadata": {},
   "source": [
    "**AIC即赤池信息准则**（Akaike Information Criterion），是衡量统计模型拟合优良性的一种标准，以下是其详细介绍：\n",
    "\n",
    "#### 定义与公式\n",
    "AIC是由日本统计学家赤池弘次在1974年提出的。它的基本思想是在衡量模型拟合程度的同时，考虑模型的复杂度，以避免出现过拟合的情况。其公式为：$AIC = 2k - 2\\ln(L)$，其中$k$是模型中参数的数量，$L$是模型的极大似然函数值。\n",
    "\n",
    "#### 作用\n",
    "- **模型选择**：在比较多个不同的统计模型对同一组数据的拟合效果时，AIC值越低的模型，被认为是越优的模型。通过比较不同模型的AIC，可以选择出最能准确描述数据，同时又不过于复杂的模型。\n",
    "- **评估拟合优度**：AIC综合考虑了模型对数据的拟合程度和模型的复杂程度。如果一个模型的AIC值较高，可能意味着该模型要么对数据的拟合效果不好，要么模型过于复杂，包含了过多不必要的参数。\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ebd6e396-e6b7-4a73-a206-67bef1c3342d",
   "metadata": {},
   "source": [
    "**变量选择**\n",
    "- 逐步回归\n",
    "    - forward stepwise regression\n",
    "    - backward stepwise regression\n",
    "    - stepwise stepwise regresson\n",
    "- 全子集回归"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "id": "75305672-d0ed-4e0a-bb5d-a3762d8f6b90",
   "metadata": {},
   "outputs": [],
   "source": [
    "library(MASS)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "id": "6e0a2d61-5666-435b-9bde-bb418fc8d24a",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Start:  AIC=97.75\n",
      "Murder ~ Population + Illiteracy + Income + Frost\n",
      "\n",
      "             Df Sum of Sq    RSS     AIC\n",
      "- Frost       1     0.021 289.19  95.753\n",
      "- Income      1     0.057 289.22  95.759\n",
      "<none>                    289.17  97.749\n",
      "- Population  1    39.238 328.41 102.111\n",
      "- Illiteracy  1   144.264 433.43 115.986\n",
      "\n",
      "Step:  AIC=95.75\n",
      "Murder ~ Population + Illiteracy + Income\n",
      "\n",
      "             Df Sum of Sq    RSS     AIC\n",
      "- Income      1     0.057 289.25  93.763\n",
      "<none>                    289.19  95.753\n",
      "+ Frost       1     0.021 289.17  97.749\n",
      "- Population  1    43.658 332.85 100.783\n",
      "- Illiteracy  1   236.196 525.38 123.605\n",
      "\n",
      "Step:  AIC=93.76\n",
      "Murder ~ Population + Illiteracy\n",
      "\n",
      "             Df Sum of Sq    RSS     AIC\n",
      "<none>                    289.25  93.763\n",
      "+ Income      1     0.057 289.19  95.753\n",
      "+ Frost       1     0.021 289.22  95.759\n",
      "- Population  1    48.517 337.76  99.516\n",
      "- Illiteracy  1   299.646 588.89 127.311\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "\n",
       "Call:\n",
       "lm(formula = Murder ~ Population + Illiteracy, data = states)\n",
       "\n",
       "Coefficients:\n",
       "(Intercept)   Population   Illiteracy  \n",
       "  1.6515497    0.0002242    4.0807366  \n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "stepAIC(fit,direction=\"both\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "id": "8e21545d-4b2b-4e9e-818c-bc6ec947ea9c",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Start:  AIC=97.75\n",
      "Murder ~ Population + Illiteracy + Income + Frost\n",
      "\n",
      "             Df Sum of Sq    RSS     AIC\n",
      "- Frost       1     0.021 289.19  95.753\n",
      "- Income      1     0.057 289.22  95.759\n",
      "<none>                    289.17  97.749\n",
      "- Population  1    39.238 328.41 102.111\n",
      "- Illiteracy  1   144.264 433.43 115.986\n",
      "\n",
      "Step:  AIC=95.75\n",
      "Murder ~ Population + Illiteracy + Income\n",
      "\n",
      "             Df Sum of Sq    RSS     AIC\n",
      "- Income      1     0.057 289.25  93.763\n",
      "<none>                    289.19  95.753\n",
      "- Population  1    43.658 332.85 100.783\n",
      "- Illiteracy  1   236.196 525.38 123.605\n",
      "\n",
      "Step:  AIC=93.76\n",
      "Murder ~ Population + Illiteracy\n",
      "\n",
      "             Df Sum of Sq    RSS     AIC\n",
      "<none>                    289.25  93.763\n",
      "- Population  1    48.517 337.76  99.516\n",
      "- Illiteracy  1   299.646 588.89 127.311\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "\n",
       "Call:\n",
       "lm(formula = Murder ~ Population + Illiteracy, data = states)\n",
       "\n",
       "Coefficients:\n",
       "(Intercept)   Population   Illiteracy  \n",
       "  1.6515497    0.0002242    4.0807366  \n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "stepAIC(fit)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "id": "ba4b8efb-2bdc-4b56-820c-7e7f3ed5410e",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Start:  AIC=97.75\n",
      "Murder ~ Population + Illiteracy + Income + Frost\n",
      "\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "\n",
       "Call:\n",
       "lm(formula = Murder ~ Population + Illiteracy + Income + Frost, \n",
       "    data = states)\n",
       "\n",
       "Coefficients:\n",
       "(Intercept)   Population   Illiteracy       Income        Frost  \n",
       "  1.235e+00    2.237e-04    4.143e+00    6.442e-05    5.813e-04  \n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "stepAIC(fit,direction=\"forward\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4088a080-5a0b-42a2-a0e7-6e7dfc185be8",
   "metadata": {},
   "source": [
    "全子集回归克服了逐步回归的模型不一定是最佳模型的限制"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "id": "13652bd8-a6ae-47b8-b758-60949ba6fc93",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "package 'leaps' successfully unpacked and MD5 sums checked\n",
      "\n",
      "The downloaded binary packages are in\n",
      "\tC:\\Users\\xie.xiaokang\\AppData\\Local\\Temp\\Rtmpw7HJat\\downloaded_packages\n"
     ]
    }
   ],
   "source": [
    "install.packages(\"leaps\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "id": "08e5fc1a-5891-4f21-b71e-96189e99e9a1",
   "metadata": {},
   "outputs": [],
   "source": [
    "library(leaps)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "id": "d971b75e-a06b-423f-9f96-c11550938366",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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2NVmdZgZK5B2q++7pa5OnhWYbvS36/zF9r9xpyluq89VaZVGAb+bJqBB/MM0rlO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\nSUVORK5CYII=",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 420,
       "width": 420
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "leaps <- regsubsets(Murder~Population+Illiteracy+Income+Frost,data=states,nbest=2)\n",
    "plot(leaps,scale=\"adjr2\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "id": "1fbb7fe1-ab5b-474d-9f46-0a9844d43245",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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DboQoeIugEQyQZd6BBRNwAi2aALHSLqBkAkG3ShQ0TdAIhkgy50iKgbAJFs0IUO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\nJAADEAnAAEQCMOD/AHhgncGeKp7NAAAAAElFTkSuQmCC",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 420,
       "width": 420
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "leaps <- regsubsets(Murder~Population+Illiteracy+Income+Frost,data=states,nbest=4)\n",
    "plot(leaps,scale=\"adjr2\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b95763a4-fd41-42ac-b9bb-f22dda295ff7",
   "metadata": {},
   "source": [
    "nbest是指选择特征组合后的前n个最佳模型\n",
    "\n",
    "#### 核心区别\n",
    "- **`nbest=2`**：对于每一种自变量的数量（从包含一个自变量的模型到包含全部自变量的模型），`regsubsets` 函数将根据评估指标（这里是调整后的 $R^{2}$）选择并返回 2 个最优的自变量子集组合。\n",
    "- **`nbest=4`**：对于每一种自变量的数量，将根据评估指标选择并返回 4 个最优的自变量子集组合。\n",
    "\n",
    "\n",
    "#### 具体影响\n",
    "- **示例说明**：假设你有 4 个自变量，即 `Population`、`Illiteracy`、`Income` 和 `Frost`，并且正在考虑构建回归模型来预测 `Murder`。\n",
    "\n",
    "##### 1. 当 `nbest = 2` 时：\n",
    "- **一元回归模型（使用一个自变量）**：会评估 4 个可能的一元回归模型（分别以 `Population`、`Illiteracy`、`Income`、`Frost` 作为唯一自变量），并根据调整后的 $R^{2}$ 选出 2 个最佳的一元回归模型。\n",
    "- **二元回归模型（使用两个自变量）**：会评估${4 \\choose 2}=6$ 种可能的二元回归模型（如 `Population` 和 `Illiteracy`，`Population` 和 `Income` 等组合），并根据调整后的 $R^{2}$ 选出 2 个最佳的二元回归模型。\n",
    "- **三元回归模型（使用三个自变量）**：会评估${4 \\choose 3}=4$ 种可能的三元回归模型（如 `Population`、`Illiteracy` 和 `Income` 等组合），并根据调整后的 $R^{2}$ 选出 2 个最佳的三元回归模型。\n",
    "- **四元回归模型（使用四个自变量）**：只有 1 种四元回归模型，包含所有自变量，会根据调整后的 $R^{2}$ 对其进行评估。\n",
    "\n",
    "##### 2. 当 `nbest = 4` 时：\n",
    "- **一元回归模型（使用一个自变量）**：会评估 4 个可能的一元回归模型（分别以 `Population`、`Illiteracy`、`Income`、`Frost` 作为唯一自变量），并根据调整后的 $R^{2}$ 选出 4 个最佳的一元回归模型。这可能会给出比 `nbest = 2` 更广泛的选择范围，但也可能包含一些相对次优的模型。\n",
    "- **二元回归模型（使用两个自变量）**：会评估${4 \\choose 2}=6$ 种可能的二元回归模型（如 `Population` 和 `Illiteracy`，`Population` 和 `Income` 等组合），并根据调整后的 $R^{2}$ 选出 4 个最佳的二元回归模型。这会比 `nbest = 2` 多选出两个最佳的二元回归模型，提供更多的选择。\n",
    "- **三元回归模型（使用三个自变量）**：会评估${4 \\choose 3}=4$ 种可能的三元回归模型（如 `Population`、`Illiteracy` 和 `Income` 等组合），并根据调整后的 $R^{2}$ 选出 4 个最佳的三元回归模型，为你提供更丰富的模型选择。\n",
    "- **四元回归模型（使用四个自变量）**：只有 1 种四元回归模型，包含所有自变量，会根据调整后的 $R^{2}$ 对其进行评估，与 `nbest = 2` 时的情况相同，因为只有一种四元回归模型组合。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "id": "7b60357d-9917-4e11-9931-7a99fb018abd",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "           Abbreviation\n",
      "Population            P\n",
      "Illiteracy           Il\n",
      "Income               In\n",
      "Frost                 F\n"
     ]
    },
    {
     "data": {
      "image/png": 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D8cRe0r0M8PoCNC+haLNN+fb9dDmhUpTI8/rqoqihzK+4o7myG1R3EN6fIAOiKk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\n/x9iTpcDj2rnjQAAAABJRU5ErkJggg==",
      "text/plain": [
       "Plot with title \"cp plot for all subsets regression\""
      ]
     },
     "metadata": {
      "image/png": {
       "height": 420,
       "width": 420
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "\n",
    "print(subsets(leaps,statistic=\"cp\",main=\"cp plot for all subsets regression\",legend=FALSE))\n",
    "abline(1,1,lty=2,col=\"dark red\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "id": "8946ed98-e0ec-4eaa-a2f7-4c8bbeda9954",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<style>\n",
       ".dl-inline {width: auto; margin:0; padding: 0}\n",
       ".dl-inline>dt, .dl-inline>dd {float: none; width: auto; display: inline-block}\n",
       ".dl-inline>dt::after {content: \":\\0020\"; padding-right: .5ex}\n",
       ".dl-inline>dt:not(:first-of-type) {padding-left: .5ex}\n",
       "</style><dl class=dl-inline><dt>(Intercept)</dt><dd>1.23456341116072</dd><dt>Population</dt><dd>0.000223675357302149</dd><dt>Illiteracy</dt><dd>4.14283659025461</dd><dt>Income</dt><dd>6.44247015228775e-05</dd><dt>Frost</dt><dd>0.000581305537729423</dd></dl>\n"
      ],
      "text/latex": [
       "\\begin{description*}\n",
       "\\item[(Intercept)] 1.23456341116072\n",
       "\\item[Population] 0.000223675357302149\n",
       "\\item[Illiteracy] 4.14283659025461\n",
       "\\item[Income] 6.44247015228775e-05\n",
       "\\item[Frost] 0.000581305537729423\n",
       "\\end{description*}\n"
      ],
      "text/markdown": [
       "(Intercept)\n",
       ":   1.23456341116072Population\n",
       ":   0.000223675357302149Illiteracy\n",
       ":   4.14283659025461Income\n",
       ":   6.44247015228775e-05Frost\n",
       ":   0.000581305537729423\n",
       "\n"
      ],
      "text/plain": [
       " (Intercept)   Population   Illiteracy       Income        Frost \n",
       "1.2345634112 0.0002236754 4.1428365903 0.0000644247 0.0005813055 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fit$coef"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 85,
   "id": "0f2e2355-91ee-4fd5-b649-9eb452e81256",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "package 'bootstrap' successfully unpacked and MD5 sums checked\n",
      "\n",
      "The downloaded binary packages are in\n",
      "\tC:\\Users\\xie.xiaokang\\AppData\\Local\\Temp\\Rtmpw7HJat\\downloaded_packages\n"
     ]
    }
   ],
   "source": [
    "# install.packages(\"bootstrap\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 95,
   "id": "1af94334-8c32-4bb7-b7c5-c11628b52d51",
   "metadata": {},
   "outputs": [],
   "source": [
    "shrinkage <- function(fit,k=10){\n",
    "    require(bootstrap)\n",
    "    theta.fit <- function(x,y){\n",
    "        lsfit(x,y)#lsfit 函数计算最小二乘回归拟合，它将一个响应变量（因变量）和一个或多个预测变量（自变量）进行线性回归分析。\n",
    "    } \n",
    "    theta.predict <- function(fit,x){\n",
    "        cbind(1,x)%*%fit$coef\n",
    "    }\n",
    "    x <- fit$model[,2:ncol(fit$model)]\n",
    "    y <- fit$model[,1]\n",
    "\n",
    "    results <- crossval(x,y,theta.fit,theta.predict,ngroup=k)\n",
    "    r2 <- cor(y,fit$fitted.values)^2\n",
    "    r2cv <- cor(y,results$cv.fit)^2\n",
    "    cat(\"origin r-square = \",r2,\"\\n\")\n",
    "    cat(k,\"fold cross validation r-square =\",r2cv,\"\\n\")\n",
    "    cat(\"change = \",r2-r2cv,\"\\n\")\n",
    "}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 96,
   "id": "a21ab0d3-7074-416b-9704-136dfe0341fb",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "origin r-square =  0.5669502 \n",
      "10 fold cross validation r-square = 0.4488856 \n",
      "change =  0.1180647 \n"
     ]
    }
   ],
   "source": [
    "shrinkage(fit)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 97,
   "id": "32bd7909-6236-4715-8454-f12e7e60e996",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "Call:\n",
       "lm(formula = Murder ~ Population + Illiteracy, data = states)\n",
       "\n",
       "Residuals:\n",
       "    Min      1Q  Median      3Q     Max \n",
       "-4.7652 -1.6561 -0.0898  1.4570  7.6758 \n",
       "\n",
       "Coefficients:\n",
       "             Estimate Std. Error t value Pr(>|t|)    \n",
       "(Intercept) 1.652e+00  8.101e-01   2.039  0.04713 *  \n",
       "Population  2.242e-04  7.984e-05   2.808  0.00724 ** \n",
       "Illiteracy  4.081e+00  5.848e-01   6.978 8.83e-09 ***\n",
       "---\n",
       "Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1\n",
       "\n",
       "Residual standard error: 2.481 on 47 degrees of freedom\n",
       "Multiple R-squared:  0.5668,\tAdjusted R-squared:  0.5484 \n",
       "F-statistic: 30.75 on 2 and 47 DF,  p-value: 2.893e-09\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "summary(fit2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 98,
   "id": "c31f4c6f-5025-4264-979b-7b5726630c7b",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "origin r-square =  0.5668327 \n",
      "10 fold cross validation r-square = 0.5296902 \n",
      "change =  0.03714245 \n"
     ]
    }
   ],
   "source": [
    "shrinkage(fit2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 100,
   "id": "effc8de8-cc15-4b8f-bec9-453fb12ea326",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<style>\n",
       ".dl-inline {width: auto; margin:0; padding: 0}\n",
       ".dl-inline>dt, .dl-inline>dd {float: none; width: auto; display: inline-block}\n",
       ".dl-inline>dt::after {content: \":\\0020\"; padding-right: .5ex}\n",
       ".dl-inline>dt:not(:first-of-type) {padding-left: .5ex}\n",
       "</style><dl class=dl-inline><dt>(Intercept)</dt><dd>-1.54414390233627e-16</dd><dt>Population</dt><dd>0.270509543317982</dd><dt>Illiteracy</dt><dd>0.684049551454848</dd><dt>Income</dt><dd>0.010723721188242</dd><dt>Frost</dt><dd>0.00818540700644149</dd></dl>\n"
      ],
      "text/latex": [
       "\\begin{description*}\n",
       "\\item[(Intercept)] -1.54414390233627e-16\n",
       "\\item[Population] 0.270509543317982\n",
       "\\item[Illiteracy] 0.684049551454848\n",
       "\\item[Income] 0.010723721188242\n",
       "\\item[Frost] 0.00818540700644149\n",
       "\\end{description*}\n"
      ],
      "text/markdown": [
       "(Intercept)\n",
       ":   -1.54414390233627e-16Population\n",
       ":   0.270509543317982Illiteracy\n",
       ":   0.684049551454848Income\n",
       ":   0.010723721188242Frost\n",
       ":   0.00818540700644149\n",
       "\n"
      ],
      "text/plain": [
       "  (Intercept)    Population    Illiteracy        Income         Frost \n",
       "-1.544144e-16  2.705095e-01  6.840496e-01  1.072372e-02  8.185407e-03 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "zstates <- as.data.frame(scale(states))\n",
    "zfit <- lm(Murder~Population+Illiteracy+Income+Frost,data=zstates)\n",
    "coef(zfit)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 185,
   "id": "93ad264e-9266-43e1-90d5-8ae98e6bf819",
   "metadata": {},
   "outputs": [],
   "source": [
    "relweights <- function(fit,...){\n",
    "    R <- cor(fit$model)\n",
    "    nvar <- ncol(R)\n",
    "    rxx <- R[2:nvar,2:nvar]\n",
    "    rxy <- R[2:nvar,1]\n",
    "    svd <- eigen(rxx)\n",
    "    evec <- svd$vectors\n",
    "    ev <- svd$values\n",
    "    delta <- diag(sqrt(ev))\n",
    "    lambda <- evec %*% delta %*% t(evec)\n",
    "    lambdasq <- lambda^2\n",
    "    beta <- solve(lambda) %*% rxy#solve用来求逆矩阵\n",
    "    rsquare <-colSums(beta^2)\n",
    "    rawwgt <- lambdasq %*% beta^2\n",
    "    import <- (rawwgt/rsquare)*100\n",
    "    import <- as.data.frame(import)\n",
    "    row.names(import) <- names(fit$model[2:nvar])\n",
    "    names(import) <- \"weights\"\n",
    "    print(import[c(order(import$weights)),1])\n",
    "    import <- import[order(import$weights),1,drop=FALSE]#drop要设置为FALSE,避免row.names被删掉\n",
    "    print(import)\n",
    "    dotchart(import$weights,labels=row.names(import),xlab=\"% of r^2\",pch=19,\n",
    "            main = \"relative importance of predictor variables\",\n",
    "            sub = paste(\"total r^2\",round(rsquare,3)),...)\n",
    "}"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4005590b-e638-413c-9a23-1f5a772e5f2f",
   "metadata": {},
   "source": [
    "- `svd <- eigen(rxx)`将相关系数矩阵分解为$E\\Lambda E^T$其中$E$为特征向量，$\\Lambda$为特征值的对角阵\n",
    "- `delta <- diag(sqrt(ev))`获得特征值平方根的对角阵$\\Delta = diag(\\sqrt\\lambda_1,\\sqrt\\lambda_2,\\sqrt\\lambda_3,\\sqrt\\lambda_4)$\n",
    "- `lambda <- evec %*% delta %*% t(evec)`就是$\\lambda = E\\Delta E^T$就相当于对预测变量的相关系数矩阵的特征值开平方后相关系数矩阵,它的逆就是$\\lambda^{-1} = E\\Delta^{-1}E^T$\n",
    "- `lambdasq <- lambda^2`这个很容易误认为是$\\lambda^2 = E\\Delta E^T E\\Delta E^T = E\\Delta^2E^T = rxx$，实际上是$\\lambda$矩阵的每个元素的平方也就是$\\lambda^2 =\\lambda\\times\\lambda$\n",
    "- `beta <- solve(lambda)%*%rxy`就是$\\beta = \\lambda^{-1}r_{xy}$\n",
    "- `rsquare <-colSums(beta^2)`就是$R^2 = \\sum\\limits_{k=1}^K \\beta_i^2$\n",
    "-  `rawwgt <- lambdasq %*% beta^2`没有归一化和百分比的各个特征的权重就是$raw\\_weight = \\lambda^2\\beta^2$\n",
    "-  `import <- (rawwgt/rsquare)*100`相对重要性$import = \\frac{raw\\_weight}{R^2}\\times 100$\n",
    "\n",
    "**ps:**\n",
    "\n",
    "$(ABC)^{-1} = C^{-1}B^{-1}A^{-1}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 243,
   "id": "6d438ad5-dda0-494a-b045-e855456549d5",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"dataframe\">\n",
       "<caption>A matrix: 4 × 4 of type dbl</caption>\n",
       "<tbody>\n",
       "\t<tr><td>0.953861905</td><td>0.001463686</td><td>0.014329414</td><td>0.030344995</td></tr>\n",
       "\t<tr><td>0.001463686</td><td>0.825264848</td><td>0.047859583</td><td>0.125411883</td></tr>\n",
       "\t<tr><td>0.014329414</td><td>0.047859583</td><td>0.929654340</td><td>0.008156664</td></tr>\n",
       "\t<tr><td>0.030344995</td><td>0.125411883</td><td>0.008156664</td><td>0.836086458</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "A matrix: 4 × 4 of type dbl\n",
       "\\begin{tabular}{llll}\n",
       "\t 0.953861905 & 0.001463686 & 0.014329414 & 0.030344995\\\\\n",
       "\t 0.001463686 & 0.825264848 & 0.047859583 & 0.125411883\\\\\n",
       "\t 0.014329414 & 0.047859583 & 0.929654340 & 0.008156664\\\\\n",
       "\t 0.030344995 & 0.125411883 & 0.008156664 & 0.836086458\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "A matrix: 4 × 4 of type dbl\n",
       "\n",
       "| 0.953861905 | 0.001463686 | 0.014329414 | 0.030344995 |\n",
       "| 0.001463686 | 0.825264848 | 0.047859583 | 0.125411883 |\n",
       "| 0.014329414 | 0.047859583 | 0.929654340 | 0.008156664 |\n",
       "| 0.030344995 | 0.125411883 | 0.008156664 | 0.836086458 |\n",
       "\n"
      ],
      "text/plain": [
       "     [,1]        [,2]        [,3]        [,4]       \n",
       "[1,] 0.953861905 0.001463686 0.014329414 0.030344995\n",
       "[2,] 0.001463686 0.825264848 0.047859583 0.125411883\n",
       "[3,] 0.014329414 0.047859583 0.929654340 0.008156664\n",
       "[4,] 0.030344995 0.125411883 0.008156664 0.836086458"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "lambdasq"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 241,
   "id": "e8d6b158-680e-48da-b488-9c3e0899ab31",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"dataframe\">\n",
       "<caption>A matrix: 4 × 1 of type dbl</caption>\n",
       "<tbody>\n",
       "\t<tr><td>0.08422982</td></tr>\n",
       "\t<tr><td>0.39253110</td></tr>\n",
       "\t<tr><td>0.01127589</td></tr>\n",
       "\t<tr><td>0.07891344</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "A matrix: 4 × 1 of type dbl\n",
       "\\begin{tabular}{l}\n",
       "\t 0.08422982\\\\\n",
       "\t 0.39253110\\\\\n",
       "\t 0.01127589\\\\\n",
       "\t 0.07891344\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "A matrix: 4 × 1 of type dbl\n",
       "\n",
       "| 0.08422982 |\n",
       "| 0.39253110 |\n",
       "| 0.01127589 |\n",
       "| 0.07891344 |\n",
       "\n"
      ],
      "text/plain": [
       "     [,1]      \n",
       "[1,] 0.08422982\n",
       "[2,] 0.39253110\n",
       "[3,] 0.01127589\n",
       "[4,] 0.07891344"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "beta^2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 237,
   "id": "ea686aa6-2605-4f02-be31-f83252188db3",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "Call:\n",
       "lm(formula = Murder ~ Population + Illiteracy + Income + Frost, \n",
       "    data = states)\n",
       "\n",
       "Residuals:\n",
       "    Min      1Q  Median      3Q     Max \n",
       "-4.7960 -1.6495 -0.0811  1.4815  7.6210 \n",
       "\n",
       "Coefficients:\n",
       "             Estimate Std. Error t value Pr(>|t|)    \n",
       "(Intercept) 1.235e+00  3.866e+00   0.319   0.7510    \n",
       "Population  2.237e-04  9.052e-05   2.471   0.0173 *  \n",
       "Illiteracy  4.143e+00  8.744e-01   4.738 2.19e-05 ***\n",
       "Income      6.442e-05  6.837e-04   0.094   0.9253    \n",
       "Frost       5.813e-04  1.005e-02   0.058   0.9541    \n",
       "---\n",
       "Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1\n",
       "\n",
       "Residual standard error: 2.535 on 45 degrees of freedom\n",
       "Multiple R-squared:  0.567,\tAdjusted R-squared:  0.5285 \n",
       "F-statistic: 14.73 on 4 and 45 DF,  p-value: 9.133e-08\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "summary(fit)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 219,
   "id": "d0824ac8-accf-4909-b314-af495293527d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"dataframe\">\n",
       "<caption>A matrix: 4 × 4 of type dbl</caption>\n",
       "<tbody>\n",
       "\t<tr><td>0.04081629</td><td>0.585048343</td><td>0.35029945</td><td>0.02383592</td></tr>\n",
       "\t<tr><td>0.41501980</td><td>0.008614911</td><td>0.08045607</td><td>0.49590922</td></tr>\n",
       "\t<tr><td>0.15935241</td><td>0.353491060</td><td>0.45792215</td><td>0.02923438</td></tr>\n",
       "\t<tr><td>0.38481150</td><td>0.052845687</td><td>0.11132234</td><td>0.45102047</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "A matrix: 4 × 4 of type dbl\n",
       "\\begin{tabular}{llll}\n",
       "\t 0.04081629 & 0.585048343 & 0.35029945 & 0.02383592\\\\\n",
       "\t 0.41501980 & 0.008614911 & 0.08045607 & 0.49590922\\\\\n",
       "\t 0.15935241 & 0.353491060 & 0.45792215 & 0.02923438\\\\\n",
       "\t 0.38481150 & 0.052845687 & 0.11132234 & 0.45102047\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "A matrix: 4 × 4 of type dbl\n",
       "\n",
       "| 0.04081629 | 0.585048343 | 0.35029945 | 0.02383592 |\n",
       "| 0.41501980 | 0.008614911 | 0.08045607 | 0.49590922 |\n",
       "| 0.15935241 | 0.353491060 | 0.45792215 | 0.02923438 |\n",
       "| 0.38481150 | 0.052845687 | 0.11132234 | 0.45102047 |\n",
       "\n"
      ],
      "text/plain": [
       "     [,1]       [,2]        [,3]       [,4]      \n",
       "[1,] 0.04081629 0.585048343 0.35029945 0.02383592\n",
       "[2,] 0.41501980 0.008614911 0.08045607 0.49590922\n",
       "[3,] 0.15935241 0.353491060 0.45792215 0.02923438\n",
       "[4,] 0.38481150 0.052845687 0.11132234 0.45102047"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "evec^2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 186,
   "id": "0e5a7741-e96a-4e2e-a539-f98fe3693025",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1]  5.488962 14.723401 20.787442 59.000195\n",
      "             weights\n",
      "Income      5.488962\n",
      "Population 14.723401\n",
      "Frost      20.787442\n",
      "Illiteracy 59.000195\n"
     ]
    },
    {
     "data": {
      "image/png": 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WomIZXPkW6b+//1kPCrmYTU2Gp34/8p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",
      "text/plain": [
       "Plot with title \"relative importance of predictor variables\""
      ]
     },
     "metadata": {
      "image/png": {
       "height": 420,
       "width": 420
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "relweights(zfit,col=\"blue\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "892dda43-3201-426d-97d7-83a77c122c1a",
   "metadata": {},
   "outputs": [],
   "source": [
    "# relweights <- function(fit,...){\n",
    "#     R <- cor(fit$model)\n",
    "#     nvar <- ncol(R)\n",
    "#     rxx <- R[2:nvar,2:nvar]\n",
    "#     rxy <- R[2:nvar,1]\n",
    "#     svd <- eigen(rxx)\n",
    "#     evec <- svd$vectors\n",
    "#     ev <- svd$values\n",
    "#     delta <- diag(sqrt(ev))\n",
    "#     lambda <- evec %*% delta %*% t(evec)\n",
    "#     lambdasq <- lambda^2\n",
    "#     beta <- solve(lambda) %*% rxy\n",
    "#     rsquare <-colSums(beta^2)\n",
    "#     rawwgt <- lambdasq %*% beta^2\n",
    "#     import <- (rawwgt/rsquare)*100\n",
    "#     import <- as.data.frame(import)\n",
    "#     row.names(import) <- names(fit$model[2:nvar])\n",
    "#     names(import) <- \"weights\"\n",
    "#     print(import[c(order(import$weights)),1])\n",
    "#     import <- import[c(order(import$weights)),1,drop=FALSE]\n",
    "#     return import\n",
    "# }"
   ]
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    "以下是对多元线性回归中自变量相对权重衡量自变量对 $R^{2}$ 重要性及其相关计算公式意义的详细解释：\n",
    "\n",
    "#### 1. 整体思路\n",
    "- 在多元线性回归中，$R^{2}$ 表示因变量的变异中可以被自变量解释的比例。相对权重的目的是将 $R^{2}$ 分解为每个自变量的贡献，以了解每个自变量在解释因变量变异中的相对重要性。它考虑了自变量之间的相关性，避免了仅使用原始回归系数或标准化回归系数时的局限性。\n",
    "\n",
    "\n",
    "#### 2. 关键步骤及公式意义\n",
    "\n",
    "##### 2.1 相关矩阵计算\n",
    "- **$R = cor(fit\\$model)$**：\n",
    "    - 计算包含因变量和自变量的模型矩阵的相关矩阵。这个矩阵反映了变量之间的线性关系强度。对于因变量 $y$ 和自变量 $x_1, x_2, \\cdots, x_p$，$R$ 中的元素 $r_{ij}$ 表示变量 $i$ 和变量 $j$ 的相关系数。\n",
    "    - 它为后续的特征分解和计算提供了基础，因为相关矩阵包含了变量间的相关结构信息，而这种结构会影响每个自变量对因变量的贡献。\n",
    "\n",
    "\n",
    "##### 2.2 提取相关部分\n",
    "- **$nvar = ncol(R)$**：获取变量的数量。\n",
    "- **$r_{xx}=R[2:nvar, 2:nvar]$**：提取自变量之间的相关矩阵，即去掉因变量所在的行和列。这个子矩阵 $r_{xx}$ 描述了自变量之间的相互关系，对于后续计算至关重要，因为自变量间的相关性会影响它们对因变量解释的相对重要性。\n",
    "- **$r_{xy}=R[2:nvar, 1]$**：提取自变量与因变量的相关向量，它表示每个自变量与因变量的相关性，是后续计算的重要部分。\n",
    "\n",
    "\n",
    "##### 2.3 特征分解\n",
    "- **$svd = eigen(r_{xx})$**:\n",
    "    - 对自变量相关矩阵 $r_{xx}$ 进行特征分解，得到特征向量 $evec$ 和特征值 $ev$。特征分解将矩阵表示为 $r_{xx}=evec\\times diag(ev)\\times t(evec)$。\n",
    "    - 特征向量 $evec$ 代表了自变量的新的正交基，特征值 $ev$ 表示在这些新基上的方差。较大的特征值对应的特征向量表示自变量空间中解释方差较大的方向，而这些方向会影响自变量对因变量解释的重要性。\n",
    "\n",
    "\n",
    "##### 2.4 计算 lambda 矩阵\n",
    "- **$\\delta = diag(\\sqrt{ev})$**:\n",
    "    - 计算特征值的平方根对角矩阵。特征值的平方根在后续计算中起到缩放特征向量的作用，以构建新的矩阵$\\lambda$。\n",
    "- **$\\lambda = evec\\,\\%\\ast\\%\\,\\delta\\,\\%\\ast\\%\\,t(evec)$**:\n",
    "    - 这个矩阵 $\\lambda$ 是根据特征分解得到的，它将原始的自变量相关矩阵 $r_{xx}$ 进行了变换。$\\lambda$ 矩阵综合了特征向量和特征值的信息，为后续求解 $\\beta$ 提供了一个新的框架，考虑了自变量之间的相关性。\n",
    "\n",
    "\n",
    "##### 2.5 求解 beta 并计算 $R^{2}$\n",
    "- **$\\beta = solve(\\lambda)\\,\\%\\ast\\%\\,r_{xy}$**:\n",
    "    - 求解方程 $\\lambda\\beta = r_{xy}$ 得到 $\\beta$。这里的 $\\beta$ 不同于普通回归中的回归系数，它是在考虑自变量相关性的情况下，通过特征分解和变换后的矩阵 $\\lambda$ 得到的。它表示在新的基下，自变量对因变量的效应。\n",
    "- **$R^{2}=colSums(\\beta^{2})$**:\n",
    "    - 计算 $R^{2}$ 为 $\\beta$ 的平方和。这是因为在多元线性回归中，$R^{2}$ 可以表示为回归系数的平方和，在这个新的框架下，$\\beta$ 的平方和表示了模型的解释能力。\n",
    "\n",
    "\n",
    "##### 2.6 计算相对权重\n",
    "- **$rawwgt=\\lambda^{2}\\,\\%\\ast\\%\\,\\beta^{2}$**:\n",
    "    - 计算原始权重。这里的 $\\lambda^{2}\\,\\%\\ast\\%\\,\\beta^{2}$ 是将变换后的矩阵 $\\lambda$ 的平方与 $\\beta$ 的平方相乘，得到的是每个自变量在新的分解下的权重贡献。\n",
    "- **$import=(rawwgt/R^{2})\\times 100$**:\n",
    "    - 计算相对权重，将原始权重标准化为对 $R^{2}$ 的百分比贡献。通过将原始权重除以总 $R^{2}$ 并乘以 $100$，得到每个自变量在解释因变量变异中的相对重要性。这使得相对权重之和为 $100\\%$，可以直观地看出每个自变量对 $R^{2}$ 的贡献比例。\n",
    "\n",
    "\n",
    "#### 3. 为什么这些计算可以衡量自变量的重要性\n",
    "- **考虑相关性**：传统的回归系数仅反映了自变量对因变量的直接效应，但在自变量相关时，这种直接效应会受到其他自变量的影响。相对权重通过特征分解，将自变量空间重新投影到新的正交基上，考虑了自变量之间的相关性，更全面地反映了自变量对因变量的综合效应。\n",
    "- **分解 $R^{2}$：** 通过一系列计算将 $R^{2}$ 分解为每个自变量的贡献，能够清晰地展示每个自变量在解释因变量变异中的相对作用。这有助于我们理解在考虑了所有自变量的共同作用和它们之间的相互关系后，哪个自变量对解释因变量变异最为重要。\n",
    "\n",
    "\n",
    "#### 4. 注意事项\n",
    "- **多重共线性影响**：如果自变量之间存在高度多重共线性，特征值会有接近零的值，导致计算的 $\\lambda$ 矩阵和 $\\beta$ 可能不稳定，从而影响相对权重的计算和解释。\n",
    "- **模型假设的重要性**：相对权重计算基于多元线性回归的假设，如误差项的独立性、同方差性和正态性。如果这些假设不成立，结果的解释可能会受到影响。\n",
    "\n",
    "\n",
    "总之，相对权重通过复杂的矩阵运算和特征分解，在考虑自变量相关性的情况下将 $R^{2}$ 分解为各个自变量的贡献，为评估自变量在多元线性回归中的重要性提供了一种更全面的方法，但在使用时需要注意数据的特性和模型的假设。"
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